Teachers’ Self-efficacy Beliefs for Teaching Math: Relations with Teacher and Student Outcomes

Journal Pre-proofs Teachers’ Self-efficacy Beliefs for Teaching Math: Relations with Teacher and Student Outcomes Harsha N. Perera, Jennifer E. John PII: DOI: Reference:

S0361-476X(20)30007-2 https://doi.org/10.1016/j.cedpsych.2020.101842 YCEPS 101842

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Contemporary Educational Psychology

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9 February 2019 20 January 2020 21 January 2020

Please cite this article as: Perera, H.N., John, J.E., Teachers’ Self-efficacy Beliefs for Teaching Math: Relations with Teacher and Student Outcomes, Contemporary Educational Psychology (2020), doi: https://doi.org/10.1016/ j.cedpsych.2020.101842

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Teachers’ Self-efficacy Beliefs for Teaching Math: Relations with Teacher and Student Outcomes Harsha N. Perera Department of Educational Psychology and Higher Education, University of Nevada Las Vegas & Jennifer E. John Department of Psychology, University of Nevada Las Vegas

Harsha N. Perera, Department of Educational Psychology and Higher Education, College of Education, University of Nevada, Las Vegas; Jennifer John, Department of Psychology, University of Nevada Las Vegas. Correspondence concerning this article should be addressed to Harsha N. Perera, Department of Educational Psychology and Higher Education, College of Education, University of Nevada, Las Vegas; Ph : 1 (702) 895-1110: E: [email protected]

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Teachers’ Self-efficacy Beliefs for Teaching Math: Relations with Teacher and Student Outcomes

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Abstract Social cognitive theory posits that teacher self-efficacy beliefs should be related to not only their own well-being outcomes but also classroom processes and student outcomes in the general ecology of the classroom environment. However, little research has directly examined the associations of teachers’ self-efficacy beliefs with these teacher and student-level outcomes simultaneously. The present study proposes and tests an integrative model of the relations of teachers’ self-efficacy for teaching math with their job satisfaction and student math achievement both directly and indirectly via interaction quality as a critical dimensions of the quality of classroom processes. Additionally, student level relational and motivational predictors of math achievement, including individual perceptions of student-teacher interaction quality and math self-concept, are included per the ecology of the classroom environment. Based on data from over 6000 4th grade students and 450 teachers, results of multilevel structural equation modeling revealed that teachers’ self-efficacy beliefs for teaching math were positively associated with teachers’ job satisfaction and class averages of math achievement and interaction quality. At the individual student level, individual levels of math self-concept were positively associated with math achievement, and individual perceptions of interaction quality were positively associated with math self-concept. However, a negative association of residualized interindividual perceptions of interaction quality with math achievement was observed. Keywords: teacher self-efficacy; student achievement; standardized achievement; teacher support; job satisfaction; social cognitive; multilevel SEM

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Research on teacher self-efficacy has rapidly increased over the past three decades (Zee & Koomen, 2016). This attention has been attributed to an increased policy focus on two related issues: (a) enhancing job satisfaction, engagement, and commitment in teachers to reduce rates of teacher attrition (McIlveen & Perera, 2018; McIlveen, Perera, Baguely, Van Rensburg, Ganguly, Jasman, & Veskova, 2019); and (b) maximizing teacher effectiveness, typically conceived of as the effects of in-class teacher beliefs and behaviors on student achievement (Klassen & Tze, 2014). Accumulating evidence shows that teacher self-efficacy is associated with a range of favorable (a) cognitive-affective teacher outcomes (e.g., higher job satisfaction, lower burnout), (b) classroom processes (e.g., better instructional management and support for students), and (c) student outcomes (e.g., greater student motivation and achievement), albeit to a lesser extent (see Zee & Koomen, 2016 for a review). However, these literatures on teacher versus student outcomes have largely developed independently of one another, limiting understanding of the role that teacher self-efficacy simultaneously plays at these different levels of classroom ecology. The ecology of the classroom environment, where teachers’ motivational beliefs may have implications for both their own and students’ outcomes, necessitates the examination of integrative models that simultaneously incorporate both teacher and student data. Indeed, as Zee and Koomen (2016) note, such integrative models are required to better elucidate the relations of teacher self-efficacy with teacher outcomes, classroom processes, and student achievement. Accordingly, drawing on social cognitive perspectives, the present study proposed and tested an integrative motivational model of the relations of teachers’ self-efficacy with educational outcomes at both teacher and student levels of classroom ecology with Australian primary school students. Specifically, the study aimed to test a model of the relations of teachers’

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self-efficacy beliefs for teaching math with their job satisfaction and classroom levels of math achievement both directly and indirectly via classroom interaction quality. In addition, individual student level relational and motivational predictors of math achievement were integrated, including individual perceptions of interaction quality and math self-concept. We examined these proposed relations using multilevel structural equation models (SEM), (Marsh et al., 2009). In doing so, we aimed to redress limitations of some previous work on teacher-self-efficacy-student outcomes relations that relied on suboptimal manifest (dis)aggregation procedures in the treatment of multilevel data with single-level models (Caprara, Barbaranelli, Steca, & Malone, 2006; Lumpe, Czerniak, Haney, Beltyukova, 2012; Mojavezi & Poodineh Tamiz, 2012), which disallowed the decomposition of individual versus teacher-level components of these relations and may have confounded these relations (Hox et al., 2018). We investigated the model in the domain of the primary-school math learning environment as a critical context in which student achievement and the evaluation of teacher effectiveness has assumed prominence (Blazar, 2015; Grigg, Perera, McIlveen, & Svetleff, 2018) Indeed, in the context of increasing teacher accountability in Australia for students’ standardized math achievement as early as grade three, there is increasing pressure on Australian primary school teachers at this level to maximize the performance of students on national math achievement testing programs (Cranley & Hine, 2016; Thompson, 2013; White & Anderson, 2011). Theoretical Background Teacher self-efficacy is theoretically predicated on Bandura’s (1986) social cognitive theory (Tschannen-Moran, Woolfolk Hoy, & Hoy, 1998). From this standpoint, teacher selfefficacy refers to self-referent judgments of capability to organize and execute the actions required to successfully perform teaching tasks. This perspective subsumes both self-perceptions

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of personal teaching capability and judgments about the requirements of domain-specific teaching tasks, including appraisals of external constraints and resources, into the teacher selfefficacy content domain (Perera, Calkins, & Part, 2019). Importantly, social cognitive theory recognizes the centrality of “person-in-context” in defining the self-efficacy construct (Zee, Koomen, Jellesma, Geerling, & de Jong, 2016). Consistent with this “person-in-context” viewpoint, domain specificity is considered to be central to efficacy judgments (Bandura, 1997, 2006; Tschannen-Moran et al., 1998), necessitating the identification of the specific teaching domain under scrutiny (Zee et al., 2016). In the present study, we are concerned with the math teaching and learning domain. In this domain, math teacher self-efficacy refers to self-referent judgments of capability to organize and execute actions required to successfully teach math, including confidence beliefs related to implementing instructional routines in math and engaging students in the math learning process. Zee and Koomen (2016) proposed a heuristic process model of teacher self-efficacy, based on Woolfolk-Hoy and colleagues’ framework of teacher self-efficacy effects (WoolkfolkHoy et al., 2009), the classroom quality framework (Pianta, La Paro, & Hamre, 2008), and social cognitive theory (Bandura, 1997). This model can be used to organize and foster theorization of links between teacher self-efficacy and educational outcomes at multiple levels of classroom ecology (see Figure 1). Key processes in this model implicate pathways linking teacher selfefficacy with the quality of classroom processes (e.g., instructional support and organization), students’ academic outcomes (e.g., academic achievement), and teachers’ satisfaction and wellbeing outcomes (e.g., job satisfaction, well-being, job commitment). These pathways suggest that greater satisfaction and well-being, better student academic outcomes, and higher quality of classroom processes are expected for teachers who believe that they are capable of successfully

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carrying out their teaching tasks. Furthermore, in line with this model, greater quality of classroom processes, such as the provision of instructional and emotional support and better classroom organization, in turn, is expected to lead to more favorable teacher and student outcomes (Zee & Koomen, 2016). Figure 1 about here Although studies have variously examined the relations of teacher self-efficacy with teachers’ satisfaction and well-being outcomes (e.g., Granziera & Perera, 2019; Klassen & Chiu, 2010; McLennan, McIlveen, & Perera, 2017), classroom processes (e.g., instructional support, classroom organization; e.g., Holzberger, Philipp, & Kunter, 2013; Nie, Tan, Liau, Lau, & Chua, 2013; Wolters & Daugherty, 2007), and students’ academic outcomes, including achievement (Caprara et al., 2006; Goddard, Hoy, & Hoy, 2000), little research has been conducted with respect to the simultaneous associations of teacher self-efficacy with these outcomes in an integrative model, as well as the relations of these classroom processes with student achievement, which would better reflect the ecology of the classroom environment. Furthermore, as Zee and Koomen (2016) note, prior efforts to examine the relations of teachers’ self-efficacy with students’ outcomes have relied on teacher reports of both teacher self-efficacy and student outcomes or used suboptimal manifest aggregation or disaggregation procedures in handling multilevel data with single-level models. For instance, some prior work has aggregated studentlevel achievement at the class and school level in single-level models whereas other work has disaggregated teacher self-efficacy data at the student level (e.g., assigning the same teacher mean self-efficacy score to each student that shares that teacher) (Caprara et al., 2006; Lumpe et al., 2012). Besides the implications of these approaches for statistical power and standard error computation (Hox et al., 2018), such approaches may lead to ecological fallacies in interpretation

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and preclude the appropriate decomposition of individual versus teacher components of these relations (Hox & Kreft, 1994; Robinson, 1950). An important extension of existing work would, then, involve the examination of relations of teachers’ self-efficacy with their work satisfaction, classroom processes, and student achievement, as well as the relations of classroom processes with teacher satisfaction and achievement, in an integrative multilevel SEM framework, which not only disentangles individual and teacher-level components of these relations but also controls for measurement and sampling error. Teacher-Level Relations Teacher Self-Efficacy and Achievement. The heuristic model of teacher self-efficacy proposed by Zee and Koomen (2016) links teachers’ self-efficacy beliefs to students’ achievement outcomes. This proposed pathway is predicated on Woolfolk-Hoy and colleagues’ (2009) process-oriented model of teacher self-efficacy, which posits that higher levels of student achievement are expected in classes with teachers who believe that they are capable of performing the tasks required to achieve desired educational outcomes. Teachers’ self-efficacy judgments influence the decisions they make about choosing, investing effort in, and persisting with goal-related activities (Woolfok Hoy et al., 2009; Bandura, 1997), which may lead to a more favorable instructional environment that promotes students’ mastery of subject content. Indeed, research shows that teachers with high self-efficacy use more student-centered constructivist instructional approaches (Nie et al., 2013), better establish mastery goal classroom structures (Cho & Shim, 2013; Deemer, 2004; Wolters & Daugherty, 2007), and provide greater learning support (Holzberger et al., 2013), all of which may foster student achievement. Consistent with this view, teacher self-efficacy has been shown to be positively-related with students’ achievement (Caprara et al., 2006), including in the math domain (Midgley et al., 1989;

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Throndsen & Turno, 2013). This relation appears to be stronger for elementary students than at the middle and high school levels, which may be explained by the relatively greater time that elementary students spend with a single teacher than their middle and high school counterparts (Zee & Koomen, 2016). Importantly, the associations of teachers’ self-efficacy beliefs with students’ outcome, such as achievement, are likely to be mediated by teaching processes (Zee & Koomen, 2016), but few studies have considered these mediational pathways. We raise this possibility of mediation of the teacher-self-efficacy-student achievement association below. Teacher Self-Efficacy and Relations with Work Satisfaction. From a social cognitive standpoint (e.g., Lent & Brown, 2006), work satisfaction is expected of those who possess higher levels of self-efficacy. Work satisfaction is a positive cognitive-affective state resulting from a favorable evaluation of work-related experiences (Perera, Vosicka, Granizera, & McIlveen, 2018). Self-efficacy beliefs impact how individuals perceive opportunities and barriers in the environment and influence the decisions people make about choosing, investing effort in, and persisting with goal-related activities. For self-efficacious teachers, beliefs that they can plan, organize, and execute the tasks required to attain desired levels of teaching performance may foster goal-related performance accomplishments with intrinsic and extrinsic rewards that are personally satisfying (Caprara et al., 2003; Granziera & Perera, 2019; Perera, Granziera, & McIlveen, 2018). Consistent with this view, prior work consistently shows that teachers with high levels of self-efficacy experience greater work satisfaction. These relations have been found across multiple educational levels (Caprara et al., 2003; Klassen & Chiu, 2010) and countries (Granziera & Perera, 2019; Klassen et al., 2009; Salanova, Llorens, & Schaufeli, 2012). Teacher Self-Efficacy and Quality of Interaction. From the perspective of social cognitive process models of teacher self-efficacy (Tschannen-Moran & Woolfolk Hoy, 2001;

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Woolfolk Hoy et al., 2009; Zee & Koomen, 2016), students who have more efficacious teachers are expected to experience better quality of student-teacher interactions. The quality of studentteacher interactions is an integral component of effective classroom processes (Zee & Koomen, 2016) and has been conceptualized as global quality dimension that can be described with respect to three domains: emotional, organizational, and instructional support (Rimm-Kauffman et al., 2015). Emotional support refers to teachers’ connection with and responsiveness to students, regard for student perspectives, and awareness of students’ needs and individual differences. Organizational support refers to the provision of clear expectations, use of a variety of instructional approaches to engage students in learning, and use of efficient teaching routines in the classroom. Instructional support refers to the presence of feedback loops in teacher-student communication, encouragement of students’ responses, and opportunities to engage in higherorder thinking (Pianta & Hamre, 2009). In the present study, we focus on students’ perceptions of these supports constituting the perceived quality of student-teacher interactions. For teachers with a strong sense of efficacy, beliefs that they can plan and execute specific teaching-related tasks to attain desired goals influence their teaching behaviors (Skaalvik & Skaalvik, 2007; Tschannen-Morian & Woolfolk Hoy, 2001), including their instructional support and classroom organizational practices. For instance, research has shown that efficacious teachers are more organized and manage classrooms better (Künsting et al., 2016; Tsouloupas, Carson, Matthews, Grawitch, & Barber, 2010), tend to use more learner-centered strategies that support student engagement and learning (Künsting, Neuber, & Lipowsky, 2016; Nie et al., 2013), and are more industrious in correcting student misunderstandings and providing feedback on student performance (Ashton & Webb, 1986; Gibson & Dembo, 1984). Accordingly, teachers with a strong sense of self-efficacy may be more likely to provide sufficient levels of support and

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organization, such that students perceive better student-teacher interaction quality (Woolfolk Hoy et al., 2009). Following previous work (Lazarides et al., 2019), we represented students’ perceived quality of interaction as a single factor reflecting perceptions of global quality of interaction (Pianta & Hamre, 2009). Interaction Quality, Work Satisfaction, and Achievement. Per Zee and Koomen’s (2016) heuristic model, interaction quality is expected to be associated with teachers’ well-being outcomes, such as work satisfaction, and student achievement outcomes. Indeed, quality of interaction with students has been theorized to be an important source of teachers’ well-being outcomes (Milatz, Lüftenegger, & Schober, 2015). From a theoretical standpoint, teachers’ emotional support of students, as a core component of interaction quality, may be driven by their basic psychological need for relatedness with students in their classes, which when fulfilled, may be a source of satisfaction (Split, Koomen, & Thijs, 2011). Moreover, supportive instructional provisions and optimal classroom organization may heighten class engagement outcomes that may be viewed as a goal-related accomplishment that is personally satisfying. Consistently, prior work shows positive links of higher quality of interaction between teachers and students with greater satisfaction (Veldman van Tartwikjm, Brekelmans, & Wubbels, 2013) and related constructs (e.g., lower burnout) (Milatz et al., 2015). For achievement, better interaction quality at the teacher level (i.e., teacher support and organization provided to all students taught by a teacher) may foster greater class levels of achievement (Hamre & Pianta, 2001) by enhancing greater class engagement (Hamre & Pianta, 2005). These associations point to the possibility of indirect relations of teacher self-efficacy with satisfaction and class achievement via class level quality of interaction. Indeed, per Zee and Koomen’s (2016) framework, such indirect relations are expected. As Zee and Koomen (2016)

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note, teacher self-efficacy—as a person characteristic that drives energetic investments and goaldirected activity—may influence student and teacher outcomes via practices that characterize the quality of the classroom environment. There is some evidence for these indirect relations. For instance, Guo et al. (2012) obtained some evidence for indirect associations of teacher selfefficacy with literacy achievement via learning support. There is also evidence that teacher selfefficacy promotes better teacher well-being outcomes via fewer classroom disturbances as an index of interaction quality at the class level (Dicke et al., 2014; see also Martin et al., 2012). Accordingly, we investigated the possibility that teacher self-efficacy may indirectly associate with teachers’ satisfaction and class-average math achievement via class levels of interaction quality. Student-Level Relations Perceptions of Interaction Quality and Achievement. Beyond relations at the teacher level, social cognitive models hold that student-level relational and motivational factors may also be associated with student achievement at the individual level within the classroom ecology (Lent, Brown, & Hackett, 2002). One salient factor may be students’ perceptions of teacherstudent interaction quality (Klem & Connell, 2004). Students who perceive that their teachers are instructionally and emotionally supportive are more likely to experience greater engagement in school and class activities (Hamre & Pianta, 2005; Hughes, Luo, Kwok, & Loyd, 2008; Klem & Connell, 2004; Ryan & Patrick, 2001) and tend to show both higher academic self-efficacy (Ryan & Patrick, 2001) and expectancies for academic success (Goodenow, 1993), all of which may foster greater achievement. In line with these views, prior work shows that students who perceive that they are supported by their teachers, including the provision of ongoing feedback (Matsumura, Patthey-Chavez et al., 2002) and meaningful and relevant instructional activities

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(Pianta, Belsky, Vandergrift, Hoults, & Morrison, 2008), perform better on achievement tasks (Arens & Morin, 2016; Hamre & Pianta, 2001; Hughes et al., 2008), including in the math domain (Crosnoe, Morrison et al., 2010). Self-concept and Achievement. At the individual student level, domain-specific academic self-concept is one of the most consistently examined predictors of student achievement. In the math-domain, math self-concept refers to individuals’ evaluative selfperceptions of their math competence and accomplishments (Shavelson, Hubner, & Stanton, 1976). From a theoretical standpoint (Valentine, DuBois, & Cooper, 2004), domain-specific selfconcept serves as a psychological mechanism of personal agency in academic contexts that informs achievement-related choices and the effort invested in learning (Trautwein, Lüdtke, Köller, & Baumert, 2006). In this motivational process, students may strive to achieve in domains in which they feel competent (Guay, Ratelle, Roy, & Litalien, 2010). Relatedly, students with favorable views of their academic competence may strive to attain achievement goals that are consistent with this self-image and, accordingly, may be more likely to execute achievement-related behaviors designed to maximize attainment (Valentine, et al., 2004). Consistently, domain-specific self-concept has been shown to be a consistent positive predictor of achievement (Valentine et al., 2005), including in the math domain with primary-school-aged students (Arens et al., 2017; Ganley & Lubienski, 2016; Pinxten et al., 2013). Perceptions of Interaction Quality and Self-Concept. Students’ perceptions of interaction quality may be related to their competence self-beliefs. Indeed, social-cognitive perspectives (Goetz, Pekrun, Hall, & Haag, 2006; Eccles & Wigfield, 2002; Lent et al., 2002) posit that perceived support in the academic environment (e.g., competence support, autonomy support, feedback), which is a critical component of heightened interaction quality, enhances

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students’ competence self-beliefs. As an evaluative self-perception, self-concept is formed through experience with, and the interpretation of, the school environment (Marsh & Craven, 1997). This includes supportive teaching provisions, such as internally-focused achievement feedback (Craven, Marsh, & Debus, 1991) and opportunities for inquiry learning that promote higher-order thinking and the application of skills (Jansen, Scherer, & Schroeders, 2015). Covariates Teacher-level Covariates. From a social-cognitive perspective on teacher motivation and satisfaction (Lent et al., 2002; Lent & Brown, 2006), teachers’ gender and years of experience may be associated with job-related beliefs, support provisions, and student-outcomes. Gender-role socialization experiences (Perera, 2016) as well as norms for gender-typed occupational activities may shape teachers’ perceptions of their gender roles in work and domestic settings and, by implication, inform their self-efficacy beliefs (Klassen & Chiu, 2010; Perera et a., 2019), interaction quality (Good, Sikes, & Brophy, 1973; Opdenakker & Van Damme, 2006), satisfaction (Liu & Ramsey, 2008), and, ultimately, class outcomes. Effects of teachers’ years of experience on their self-efficacy beliefs (Klassen & Chiu, 2010, Perera et al., 2019) and satisfaction (van der Ploeg & Scholte, 2003) have also be observed, which have been explained in terms of teacher career development models (e.g., Huberman, 1989). Accordingly, we included teacher gender and years of experience as covariates in the model. Student-level covariates. Social cognitive perspectives also posit roles for student gender and SES in achievement-related processes (Lent et al., 2002). Accumulating research has shown that girls and boys perform comparably in math domains and, where differences exist, these tend to be small (Else-Quest, Hyde, & Linn, 2010; Grigg et al., 2018). However, large gender differences have been found in math-related motivational beliefs. Specifically, girls

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consistently report lower math self-concepts than do boys (Eccles & Wang, 2016; Goldman & Penner, 2016; Skaalvik & Skaalvik, 2004). Gender has also been shown to associate with students’ perceptions of interaction quality, including support provisions (Reddy, Rhodes, & Mulhall, 2003). Accordingly, we accounted for the effects of student gender on math selfconcept, perceived teacher support, and achievement. Furthermore, research consistently shows that students from low SES backgrounds show more limited math skills and achievement (Claessens & Engel, 2013; Jordan et al., 2006; Klein et al., 2008; National Research Council, 2009), which has been attributed in part to issues of access to learning tools in home learning environments (Bradely et al., 2002). Accordingly, we controlled for the effects of student SES on achievement (Sirin, 2005). The Present Study The theory and available evidence reviewed is suggestive of a model in which teachers’ math teaching self-efficacy is linked with classroom levels of interaction quality. Additionally, the model indicates that teachers’ math teaching self-efficacy may be associated with their job satisfaction and classroom levels of math achievement both directly and indirectly via classroom levels of interaction quality. In addition, the literature reviewed suggests important individuallevel relations in the model in line with the ecology of the classroom environment. Specifically, individual levels of students’ perceptions of interaction quality should be associated with individual levels of students’ math self-concept and achievement, and these competence selfperceptions should be related to math achievement. The model examined is shown in Figure 2. These relations were tested using data from over 6000 students and 450 teachers. Figure 2 about here The specific hypotheses implied by this model are as follows:

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Hypothesis 1 (H1): Teachers’ self-efficacy for teaching math will be positively and directly associated with (a) their job satisfaction as well as classroom levels of (b) interaction quality and (c) math achievement. Hypothesis 2 (H2): Classroom levels of interaction quality will be positively associated with (a) teachers’ work satisfaction and (b) class math achievement. Hypothesis 3 (H3): Teachers’ math teaching self-efficacy will be positively and indirectly associated with (a) their job satisfaction and (b) classroom levels of math achievement via interaction quality. Hypothesis 4 (H4): Individual levels of students’ perceptions of interaction quality will be positively associated with individual levels of (a) math self-concept and (b) math achievement. Hypothesis 5 (H5): Individual levels of students’ math self-concept will be positively associated with individual levels of math achievement. We tested these hypotheses while accounting for the effects of teacher gender and years of experience and student gender and SES. Method Participants and Procedure The present study drew on Australian data for teachers and their students from the Grade 4 TIMSS 2015 Mathematics Assessment. Directed by the International Association for the Evaluation of Educational Achievement (IEA), TIMSS 2015 is an international large-scale study designed to provide comparative data about math and science educational achievement across participating countries and regions. The Grade 4 TIMSS 2015 Assessment comprises data from forty-nine countries and seven sub-national entities, including data from fourth grade students,

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their teachers, parents, school principals, and curriculum experts. The Grade 4 TIMSS 2015 Mathematics Assessment involved data collection in three broad areas as follows: (a) student achievement outcomes, characteristics, and self-perceptions and attitudes towards learning; (b) school, teacher and classroom contexts; and (c) national and community contexts. Following survey development and field trials, Australian participants competed the TIMSS 2015 Assessment from October to November 2014. Participating students completed a paper-based assessment, including math assessment items, administered in two sessions, with each session lasting approximately 36 minutes. Additionally, each student completed a questionnaire comprising items about socio-demographic characteristics, home environment, and selfperceptions and attitudes towards learning math and science. Participating teachers were administered a questionnaire, requiring approximately 35 minutes to complete, which contained items about socio-demographics, teacher preparation, professional development, instructional practices, self-perceptions, assessment practices, school and classroom climate, and professional development. Participating teachers were those that provided math instruction to the students surveyed. The Australian sample from the Grade 4 TIMSS 2015 Mathematics Assessment analyzed in this study comprises 452 teachers who taught 6057 fourth grade students. Teachers from all states and territories of Australia participated in the study. Participating teachers reported their age in one of six categories: “under 25” (n = 13, 2.9%); “25-29” (n = 69, 15.3%); “30-39” (n = 95, 21.0%); “40-49” (n = 107, 23.7%); “50-59” (n = 103, 22.8%); “60 or more” (n = 30, 6.6%). Thirty-five (7.7%) teachers did not report their age range. Teachers reported diverse teaching experience, ranging from one year to 46 years (M = 15.505, SD = 11.494) The majority of the teacher sample was female (n = 341, 75.4%), as is commonplace in Australian primary schools.

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Thirty-two (7.1%) teachers did not report their gender and years of teaching experience. On average, teachers taught 13.400 students. The mean age of participating students was 10.046 (SD = 0.391), and 48.5% (n = 2937) of the student sample was female. Forty-eight students (0.8%) did not report their gender. The present study used data on teachers’ self-efficacy beliefs for teaching math, job satisfaction, and socio-demographics, as well as students’ math achievement, self-concept, perceptions of interaction quality, and socio-demographics as outlined below. Measures Teacher self-efficacy. Teachers’ self-efficacy for teaching math was measured using nine items from the TIMSS 2015 Teacher Questionnaire Mathematics, which were designed to measure teachers’ perceived capabilities to organize and execute actions required to successfully teach math.1 Items were rated on a four-point Likert-type scale, ranging from 1 (Very High) to 4 (Low). All items were recoded, such that higher values reflected higher levels of teacher selfefficacy. A sample item is “in teaching mathematics to this class, how would you characterize your confidence in doing the following? Making mathematics relevant to students”. Scores generated from the measure have been shown to be reliable, and criterion-related validity evidence has been obtained (Caponera & Losito, 2016). The coefficient alpha reliability in the present sample was acceptable (α = .911).2 Teacher job satisfaction. Teachers’ job satisfaction was measured using the TIMSS 2015 Teacher Job Satisfaction Scale. The scale includes seven items, rated on a 4-point Likerttype scale ranging from 1 (Very Often) to 4 (Never or Almost Never), designed to measure

Items used to measure each construct are shown in Appendix A. Please note that model-based composite reliability estimates (i.e., coefficient omega; McDonald, 1970) for the factors are reported in the Results section based on retained measurement model solution. Composite reliability, operationalized via the Omega coefficient, should be preferred to coefficient alpha as an index of reliability as we explain below. 1 2

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teachers’ cognitive-affective evaluation of satisfaction with work. A sample item is “I am satisfied with being a teacher at this school”. All items were recoded so that higher values reflected higher levels of job satisfaction. Evidence for structural validity of the scale scores has been obtained (Martin et al., 2016), and the coefficient alpha reliability in the present sample was acceptable (α = .919). Student math self-concept. Students’ math self-concept was measured using seven items from the TIMSS 2015 Student Questionnaire, which were designed to index students’ evaluative self-perceptions of their math competence.3 Items were rated on a four-point Likerttype scale, ranging from 1 (Agree a lot) to 4 (Disagree a lot). All items were recoded, such that higher values reflected higher levels of math self-concept. A sample item is “I usually do well in mathematics”. Validity evidence, including structural and criterion-related validity evidence, for the item data has been obtained (Martin et al., 2016), and the coefficient alpha reliability for the scale scores in the current sample was acceptable (α = .840).4 Perceived Quality of Student-Teacher Interactions. Students’ perceptions of studentteacher interaction quality in math lessons were measured using 10 items from the TIMSS 2015 Student questionnaire. These items index student views of the extent to which teachers provided emotional support, instructional support, and a well-organized math learning environment. Items were answered with respect to a common prompt: “How much do you agree with these statements about your mathematics lessons?”. Sample items include “my teacher does a variety

Although TIMSS 2015 refers to the math self-concept items as the “Students Confident in Mathematics Scale”, these items do not index confidence or self-efficacy beliefs but, rather, self-concept (i.e., evaluative judgments of competence; Guo et al., 2015, 2017; Marsh et al., 2013). 4 Note that the TIMSS 2015 Students Confident in Mathematics Scale, which we used to index self-concept, contains nine items. However, we opted to use only seven of the nine items as two items indexed constructs distinct from math self-concept. The two items were “Mathematics makes me nervous” and “Mathematics makes me confused”. Both these items were taken from the Mathematics Anxiety Scale Revised and reflect math anxiety rather than math self-concept. To avoid construct contamination and confounding we elected no to include these items as indicators of math self-concept. 3

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of things to help us learn” (classroom organization), “my teachers tells me how to do better when I make a mistake” (instructional support), and “My teacher listens to what I have to say” (emotional support). Items were rated on a four-point Likert-type scale, ranging from 1 (Agree a lot) to 4 (Disagree a lot). All items were recoded so that higher values reflected higher levels of perceived student-teacher interaction quality. Validity evidence for the item data, including structural and criterion-related validity evidence, has been obtained (Martin et al., 2016), and the coefficient alpha reliability estimate in the present sample was acceptable (α = .885). Socio-demographics. In the TIMSS 2015 assessment, teachers and students reported their gender on a binary response scale. Responses were coded, such that 0 = male and 1 = female. Teachers also reported on their years of experience in the profession measured continuously. Student SES was indexed using the TIMSS 2015 Home Resources Scale. This scale was developed using IRT scaling based on responses to four educational resources items, including number of books in home, highest level of education of either parent, number of home study supports, and highest level of occupation of either parent, with higher levels on the scale indicating more resources (Mulis, Martin, Fpy, & Hooper, 2016). Math achievement. Student math achievement was measured using the TIMSS 2015 Fourth Grade Math Assessment. This assessment comprises three content domains, namely numbers (e.g., whole numbers, fractions and decimals), geometric shapes and measures (e.g., points, lines, and angles), and data display (e.g., reading and interpreting data). TIMSS 2015 used Item Response Theory (IRT) scaling and plausible values methodology to obtain students’ math achievement scores (TIMSS & PIRLS International Study Center, 2016). Plausible values are random draws from a distribution of proficiency values that reasonably represent the range of

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ability a student may plausibly have (Foy & Lin, 2015). Each student was assigned five values from a distribution of scores that could be plausibly assigned. Statistical Analyses Analyses were conducted in four phases using multilevel structural equation modeling (SEM) with students at Level 1 (L1) and teachers at Level 2 (L2). First, we assessed the L2 variability for constructs specified at both L1 and L2 (i.e., interaction quality and achievement) by computing ICC1 values. Additionally, we assessed the level of agreement among students in the ratings of these L2 constructs using the ICC2, which provides an index of the reliability of aggregated L2 measures (Marsh et al., 2012). Sufficient variability at L2 and acceptable agreement in L2 aggregates constitute important assumptions of doubly-latent multilevel SEM models.5 Second, a multilevel CFA model examined whether the a priori measurement model holds at both levels. In this model, teacher self-efficacy and job satisfaction were specified as independent clusters factors at L2. For perceived student-teacher interaction quality, student ratings on the 10 manifest indicators were specified to load onto single latent factors at L2 and L1. The L2 component of these ratings reflects the class aggregate of interaction quality based on the agreement between ratings of students who share the same teacher. Contrariwise, the L1 component reflects deviations from the classroom-average ratings given inherent group-mean centering (Morin, Marsh, Nagengast, & Scalas, 2014). Math achievement was specified as a single indicator at L1 with latent aggregation to L2 to provide an index of class average achievement. Math self-concept was specified as an independent clusters CFA factor at L1 indexed by its respective indicators.6 For this factor, we also specified a set of a priori within-

5

ICC estimates are available for only factors specified at both L1 and L2.

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construct correlated uniquenesses to account for construct-irrelevant multidimensionality due to item method effects emerging from the presence of positively and negatively keyed items (Marsh, Scalas, & Negengast, 2010; Perera, McIlveen, Burton, & Corser, 2015).7 Across the latent variables, loadings were freely estimated at both levels, excepting the first indicator-to-factor loading, which was fixed to 1.00 for identification. Factors were permitted to freely covary. Students’ gender and SES were incorporated as L1 covariates, and teacher gender and years of experience were specified as L2 covariates. To foster interpretation and reduce non-essential multicollinearity, variables, except those used at L1 only and the covariates, were standardized to have means equal to zero and unit variance (Morin et al., 2014). In addition, variables specified at L1 only, except the categorical L1 covariates, were centered on their grand means. Per TIMSS protocol, all statistical analyses were conducted with each of the five achievement plausible values separately, and the results were aggregated using Rubin’s pooling rules (Little & Rubin, 2002). In Phase 3, the multilevel CFA model was re-specified with factor loadings of constructs specified at L1 and L2 constrained to equality across these levels. Although loading invariance across L1 and L2 is not required, it has several advantages (Lüdtke, Marsh, Robitzsch, & Trautwein, 2011; Mehta & Neale, 2005; Marsh et al., 2012; Morin et al., 2014). Firstly, loading invariance equates the factor metric across L1 and L2 (Lüdtke et al., 2014), enhancing construct comparability across levels. Invariant loadings also lead to a more parsimonious model, enhancing the stability and accuracy of estimates (Lüdtke et al., 2014). Finally, the retained measurement model was respecified to reflect the a priori predictions for structural relations among the constructs. In this multilevel SEM, direct paths from teacher

For the self-concept factor, covariances between the uniqueness terms for (a) Item 1 with Items 4, 5, and 6, (b) Item 4 with Items 5 and 6, and (c) Items 5 with Item 6 were specified. 7

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self-efficacy to job satisfaction as well as classroom levels of interaction quality and achievement at L2 were specified. Direct paths from classroom interaction quality to satisfaction and achievement at L2 were also specified. In addition, teacher self-efficacy, job satisfaction, and classroom level of interaction quality and achievement were regressed on teacher gender and years of experience. At L1, individual math achievement was regressed on math-self-concept and perceived interaction quality. In addition, a path from interaction quality to math self-concept was specified. Finally, individual levels of math self-concept, interaction quality perceptions, and math achievement were regressed on student gender and math achievement was regressed on SES. Analyses were conducted using Mplus 8.0 (Muthén & Muthén, 1998-2017). Solutions were estimated using maximum likelihood estimation with the Yuan-Bentler (2000) correction to the likelihood ratio test statistic and sandwich estimator standard errors that are robust to multinon-normality and the ordered categorical nature of the observed data where there are at least four response categories (Beauducel & Herzberg, 2006). This estimation routine is operationalized via the MLR estimator in Mplus (Muthén & Muthén, 1998-2017). For model fit evaluation, an inclusive approach was used involving a consideration of fit indices and the theoretical consistency and admissibility of parameter estimates. As the χ2 can be oversensitive to minor model misspecifications given even moderate-sized samples and contains a restrictive hypothesis test (i.e., exact fit), three approximate fit indices were used: Root Mean Square Error of Approximation (RMSEA), ≤ .050 and .080 for close and reasonable fit, respectively; Comparative Fit Index (CFI); and Tucker-Lewis Index (TLI), ≥ .900 and .950 for acceptable and excellent fit, respectively. For model comparisons, we relied on changes in the CFI (ΔCFI) and RMSEA (ΔRMSEA). A decrease in the CFI and increase in the RMSEA of less than .010 and

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.015, respectively, are indicative of support for a more restrictive model (Chen, 2007; Cheung & Rensvold, 2002). Indirect associations were computed as the product of two implicated path coefficients and tested using delta method standard errors (Mackinnon, 2008). There was a generally small amount of missing data on the observed indicators. Missingness ranged from 8.8%-9.1% for indicators of teacher self-efficacy and 8.4% - 8.6% for indicators of jobs satisfaction. On student ratings, missingness ranged from 1.6%-3.9% for indicators of self-concept and 1.8%-2.6% for indicators of perceived teacher support. There were no missing data on the plausible values for the math achievement scores. Little’s (1988) MCAR test indicated that the data were inconsistent with the MCAR mechanism, χ2 (5709) = 8041.376, p < .001.Accordingly, we used full information maximum likelihood (FIML), in conjunction with the MLR estimator in Mplus, to handle the missingness. FIML has been shown to result in unbiased and more efficient estimates, even with a high level of missing observations assuming data are missing at random (Enders & Bandalos, 2001). In addition, FIML has been shown to generally outperform multiple imputation procedures in multilevel analyses (Larsen, 2011) and is especially advantageous given the presence of plausible values, which themselves require imputation-like pooling procedures. Results Preliminary Results and Measurement Models Prior to evaluating the hypotheses, we examined ICC1 and ICC2 values. In multilevel studies, ICC1 values should approximate .10 but seldom exceed .30 (Lüdtke et al., 2008, 2011; Morin et al., 2014).8 For these data, ICC1 values were reasonable for interaction quality (ICC1 = .143) and math achievement test scores (ICC1 = .341). Lower levels of L2 variability for

8

Observed sample statistics are shown in Appendix B for the observed variables.

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interaction quality relative to standardized achievement test scores is consistent with previous work (Arens & Morin, 2016). ICC2 values were also acceptable for both interaction quality (ICC2 = .689) and achievement (ICC2 = .874). Although the value for interaction quality reflects slightly lower reliability (i.e., agreement of students in ratings of L2 constructs), in doubly-latent multilevel models this measurement error is controlled via latent aggregation (Morin et al., 2014). Measurement Model Next, we examined the multi-level CFA models to determine whether the a priori measurement model provided a sufficient representation of the data and to test the invariance of the factor loadings across levels. As shown in Table 1, the test of the multilevel CFA model with no cross-level invariance restrictions resulted in an acceptable fit to the data.9 This model was compared to a multilevel CFA model in which the factor loadings for the interaction quality factor were constrained to equality across levels. This more parsimonious model resulted in an acceptable fit in absolute terms, and, notably, did not result in an appreciable degradation in fit relative to the less restrictive model. This result supports the invariance of the factor loadings for interaction quality across levels. Tables 2 and 3 show the factor correlation matrices from the final invariant measurement model for L1 and L2, respectively. At L1, interaction quality was positively and significantly associated with math self-concept and, to a lesser extent, math achievement, and math self-concept was positively associated with math achievement. At L2, teacher self-efficacy was significantly and positively associated with math achievement and work satisfaction; its association with class levels of interaction quality was non-trivial but did

In an initial run of the Multilevel CFA model (with no cross-level invariance constraints), there was a small negative residual variance (-.001) for the ASBM02E indicator of Interaction Quality at L2, which is not uncommon at the between level. This residual variance was fixed to zero. 9

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not reach significance at p <.05. The class-average interaction quality was significantly and positively associated with work satisfaction but not with class-average math achievement. The association between math achievement and work satisfaction was positive and statistically significant. Table 1 about here Table 2 about here Table 3 about her In this solution with cross-level loading invariance, teacher self-efficacy (λcs = .683-.767, M = .728), job satisfaction (λcs = .698-.893, M = .792), and interaction quality (λcs = .800-.999, M = .898) at L2 were well defined with uniformly strong and statistically significant factor loadings. Similarly, at L1, math self-concept (λcs = .293-.791, M = .615) and student perceptions of interaction quality (λcs = .593-.732, M = .675) were well-defined. From the retained measurement model, for the multi-indicator constructs, we computed composite reliability coefficients at L1 and L2 as McDonald’s (1970) omega (ω) coefficient.10 Composite reliabilities were acceptable for teacher self-efficacy (ωL2 = .911), job satisfaction (ωL2 = .923), interaction quality (ωL1 = .8876, ωL2 = .977), and math self-concept (ωL1 = .816). Structural Model From the retained multilevel CFA model with invariant loadings across levels, the structural relations were specified and tested per the hypotheses advanced. Results of the test of the structural model are reported in Table 1. The test of this multilevel SEM model resulted in an

10

McDonald’s (1970) Omega coefficient of composite reliability is given as ω = (Σ|λi|)2 / (Σ|λi|)2 + Σδii), where λi are the factor loadings and δii are the residual variances obtained from a common factor model analysis. The Omega coefficient should be preferred to the Cronbach’s alpha coefficient as it accounts for the strength of the association of each item with its corresponding latent factor, controls for item errors of measurement, and does not assume essential tau-equivalence (Sijtsma, 2009).

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acceptable fit to the data, and, notably, no appreciable decrement in fit relative to the more complex retained measurement model. Table 4 shows parameter estimates from this multilevel SEM model. Beginning with L2, consistent with H1a, math teaching self-efficacy was positively and significantly associated with job satisfaction. Additionally, in line with H1b and H1c, teachers’ self-efficacy for teaching math was positively and significantly associated with the class averages of interaction quality and math achievement, respectively. The L2 association between the class average of interaction quality with job satisfaction (H2a) and math achievement (H2b) were not statistically significant. At L1, consistent with H4a, residualized student perceptions of interaction quality was positively associated with math self-concept. However, inconsistent with H4b, interaction quality was negatively and significantly, albeit weakly, related to achievement. In line with H5, individual levels of students’ math self-concept were positively associated with math achievement. Our tests of the indirect associations were inconsistent with the hypotheses advanced. Inconsistent with H3a, teacher self-efficacy was not statistically significantly indirectly associated with job satisfaction via class levels of interaction quality (γβcs = .017, SE = .014, p = .211). Furthermore, inconsistent with H3b, teacher self-efficacy was not statistically significantly indirectly associated with class-average math achievement via class levels of interaction quality (γβcs = .013, SE = .014, p = .368). Table 4 about here Discussion Although prior research has consistently shown that teachers’ self-efficacy beliefs foster important teacher level-outcomes, such as better job satisfaction (Ganziera & Perera, 2019; Klassen & Chiu, 2010) and commitment (Klassen & Chiu, 2011), comparatively less is known

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about the simultaneous role of teacher self-efficacy beliefs in their own well-being outcomes and students’ achievement-related outcomes both directly and indirectly via quality of interaction with students (Zee & Koomen, 2016). We addressed this gap in the literature by testing an integrative social cognitive model of the relations among teachers’ math teaching self-efficacy, interaction quality in the math classroom, job satisfaction, and student math achievement. Additionally, per social cognitive perspectives and the ecology of the classroom environment, traditional student-level predictors of achievement were modeled, including students’ math selfconcept and individual perceptions of interaction quality. Tests of the model yielded support for a subset of the hypotheses advanced. We discuss the findings of the study with respect to theory and prior empirical work below. L2 Relations Resulted showed that teachers who reported greater math teaching self-efficacy were more likely to be satisfied with their jobs. This result converges with a large body of prior work showing positive relations of teacher self-efficacy with job satisfaction (Granizera & Perera, 2019; Klassen & Chiu, 2010; Skaalvik & Skaalvik, 2010). However, the association observed was somewhat smaller than the relations found in extant work (e.g., Caprara et al., 2006; Klassen et al., 2009; Klassen & Chiu, 2010). In prior work, teacher self-efficacy was measured with respect to the general teaching domain as opposed to self-efficacy for teaching a particular subject, ensuring optimal correspondence between the teacher self-efficacy domain and the satisfaction domain (i.e., teaching broadly) (Pajares, 1996). Although satisfaction with teaching in general may be influenced by teachers’ capability beliefs for teaching specific subjects, it is likely that math teaching self-efficacy beliefs are just one component of the broader motivational system of capability beliefs that influence teachers' job satisfaction.

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Teachers’ self-efficacy for teaching math was also found to be positively related to the class-average of students’ perceptions of interaction quality. This positive association may be explained by the observation that teachers with stronger teaching self-efficacy beliefs are better organized and better manage the classroom, use more learner-centered teaching strategies that are cognitively activating, and provide more timely feedback on student learning (Künsting et al., 2016; Nie et al., 2013), all of which may be perceived by students as supporting their learning. Importantly, we extend prior work on the relations of teacher self-efficacy with interaction quality, which has been based, in part, on teacher reports of both self-efficacy beliefs and interaction quality components (Künsting et al., 2016; Nie et al., 2013), by showing that the positive association holds with a more objective index of interaction quality based on the agreement of ratings of interaction quality by students who share the same teacher. Teacher self-efficacy for teaching math was also found to positively associate with the class-average of math achievement, indicating that classes taught by teachers with stronger selfefficacy beliefs for teaching math tended to have higher average math achievement. This result aligns with the heuristic model of teacher self-efficacy relations proposed by Zee and Koomen (2016), as well as the process-oriented model of teacher self-efficacy effects on which this model is predicated (Woolkfolk et al., 2009), which hold that greater achievement is expected among students who have teachers with stronger teaching self-efficacy beliefs. The result also converges with a small amount of prior work that has found positive relations of teacher self-efficacy with student achievement (Caprara et al., 2013; Midgley et al., 1989; Throndsen & Turno, 2013). Importantly, we extend this work by finding evidence for the relation based on multilevel models that appropriately decompose individual and teacher components of these relations. Furthermore, this association was found with a standardized measure of math achievement, which is not as

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susceptible to the “grading-on-the-curve” problems inherent in teacher-reported grades and better reflects students’ levels of math attainment (Marsh et al., 2014). Despite converging with social-cognitive theories and some prior work (Caprara et al., 2006), this finding of a positive association of teacher self-efficacy for teaching math with the class-average of math achievement is seemingly inconsistent with recent results from Zee, Koomen, and de Jong (2018). Zee and colleagues found that general teacher self-efficacy was not significantly associated with students’ math achievement. One reason for these discrepant results is Zee and colleagues’ use of a domain general measure of teacher self-efficacy. On the contrary, a measure of self-efficacy for teaching math was used in the present study. Selfefficacy for teaching math is conceptually more proximal to the math learning domain consistent with the theorized domain-specificity of teacher self-efficacy beliefs (Tschannen-Moran et al., 1998). The discrepancy in findings may point to the centrality of domain correspondence in investigations of the correlates of teacher self-efficacy, especially in relation to student outcomes, as has been found in other work (Pajares, 1996; Perera et al., 2019; Ross & Bruce, 2007). Although the indirect relations of teachers’ math teaching self-efficacy with student achievement and job satisfaction were consistent with the expected direction of effects, these associations were small and not significant. This is somewhat surprising as, from a conceptual standpoint, better quality of classroom processes, reflected, in part, by heightened interaction quality, is posited to be a mechanism by which teacher self-efficacy transmits its effects onto teacher and student outcomes (Zee & Koomen, 2016). One possible reason for the near-null relations is the measurement of quality of classroom processes in the present study, which centered on students’ shared perceptions of teacher-student interaction quality at L2. Teachers’

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job satisfaction may hinge more heavily on their own beliefs about interaction quality with students (Spilt, Koomen, & Thijs, 2011). Furthermore, for achievement, standardized achievement tests, which are momentary assessments that require short-term engagement, may depend more so on objective teacher practices that foreground higher-order thinking skills and the provision of process-oriented feedback (quite apart from students’ perceptions thereof), which support academic development (Hamre & Pianta. 2005; Gibson & Dembo, 1984; Guo et al., 2012; Zee & Koomen, 2016). Nevertheless, and notwithstanding our findings of direct relations of teachers’ math teaching self-efficacy with job satisfaction and achievement, it unlikely that the effects of teachers’ math teaching self-efficacy are exclusively direct. For instance, for job satisfaction, previous research has found significant mediation via perceived work engagement (Granziera & Perera, 2019) and perceived work conditions (Duffy & Lent, 2009; Lent et al., 2011). For standardized achievement, comparatively less is known, but, from a conceptual standpoint, significant mediation may be expected via observable teaching practices that support academic development (Guo et al., 2012) and future research would do well to further examine potential mediational mechanisms. L1 Relations Beyond the associations observed at L2, the present study found support for some relations among perceived interaction quality, math self-concept, and math achievement at the individual level. The present study replicated a good deal of prior work showing that students who have favorable evaluative self-perceptions of their math competence perform better in math (Ganley & Lubienski, 2016; Guo, Marsh, Parker, Morin, & Yeung, 2015; Pinxten et al., 2013). Our results also showed that students’ who perceived greater quality of interaction with their teachers were more likely to hold more favorable math self-concept beliefs. For these students,

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perceptions of greater emotional and instructional supports in the classroom, including provisions of performance feedback and praise, may be critical in the system of reinforcement that underlies the development of self-concept (Burnett, Craven, & Marsh, 1999; Marsh & Craven, 1997). However, future research is required to examine this more directly. A somewhat unexpected finding is that students’ perceptions of interaction quality at the individual level were weakly and negatively, though statistically significantly, associated with individual levels of math achievement. This result may seem somewhat unexpected as interaction quality, involving the provision of emotional support, instructional support, and a well-organized classroom environment, has been theorized to enhance motivation, engagement, and ultimately achievement (Hamre & Pianta, 2009). Notably, negative links of students’ perceptions of teacher support, constituting, in part, interaction quality, with achievement have been found in previous research (Lee & Stankov, 2013). At L2, students’ ratings of support reflect class-aggregates of interaction quality based on the agreement in ratings of students who share the same teacher. Beyond this L2 component, at L1, the construct reflects residualized inter-individual perceptions of quality (i.e., individual deviations from class-average ratings of support). It may be that, at L1, students who perceive more support from the teacher as part of their interactions are those students who are academically struggling with math and underperform. These underperforming students may draw more attention and support resources from the teachers in an effort to improve their academic position. Practice Implications and Limitations The results of this study have some tentative educational implications. Taken together, the findings of this study suggest that fostering an effective classroom environment, characterized by positive interaction quality and favorable teacher and student outcomes,

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requires a consideration of teachers’ self-efficacy beliefs (Caprara et al., 2006). In the present study, teachers’ self-efficacy beliefs were found to be associated with their work satisfaction and classroom levels of interaction quality and achievement. Teachers’ self-efficacy beliefs have also been found to be related to a host of other desirable outcomes, such as teachers’ planning and organization (Allinder, 1994), lower burnout (Egyed & Short, 2006), and commitment to the profession (Klassen & Chiu, 2010). As such, resources should be invested in professional development learning opportunities that allow for teachers’ continuing development of their capability beliefs. Participation in professional development provides teachers with an opportunity (a) to develop and strengthen skills and knowledge that may enhance teaching performance to a desired level of effectiveness, (b) to receive feedback and verbal encouragement on teaching innovations, and (c) for vicarious learning experiences as teachers observe proficient models who demonstrate new teaching routines (e.g., peer observations), all of which may foster the development of teachers’ capability beliefs (Ross & Bruce, 2007; Tchannen-Moran & McMaster, 2009). The findings also suggest that the maximization of students’ math achievement requires addressing students’ math self-concept beliefs. The results suggest that one possibility for addressing self-concept is the provision of emotional and instructional classroom support, including performance feedback and praise, which may be important in the development of competence self-perceptions (Burnett et al., 1999; Marsh & Craven, 1997; Craven et al., 1991). This is particularly true for under-achieving students who may rely more on such support provisions in the classroom. However, these implications remain tentative given the cross-sectional nature of the data and the possibility of reverse causality and reciprocal effects.

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Limitations to this research merit attention. First, data collected in the TIMSS assessment were cross-sectional, thereby precluding directionality inferences from the associations observed as well as control for prior variance in outcomes, which weakens the robustness of inferences relating to predictive effects. Indeed, one important condition that needs to be satisfied before causal relations between two variables can be reasonably inferred is that there is isolation (i.e., the statistical associations between variables hold after controlling for other variables that may affect the outcomes) (Kline, 2012). However, to the extent that levels of the outcome at time t - 1 affect levels of the outcome at time t (i.e., the autoregressive effect is a tenable effect), and such prior variance is not controlled as with cross-sectional data, isolation cannot be achieved. A related concern with the cross-sectional data is that it precludes investigation of theoretically meaningful reciprocal relations among the constructs. For instance, from a social cognitive standpoint (Woolfolk Hoy et al., 2009; Zee & Koomen, 2016), teacher self-efficacy beliefs and students’ achievement may be expected to be reciprocally related over time. Self-efficacy beliefs may promote better achievement, and, in turn, better student achievement may be interpreted by teachers as indicative of their own attainment and serve as a mastery experience that informs future teaching self-efficacy beliefs. Future research is required to examine these relations with longitudinal data to provide a more robust test of the relations proposed. Another concern is that, though social cognitive perspectives were used to theorize the constructs and relations among the constructs in the present study, several other frameworks may be used to conceptually frame the study and may provide additional insight into the relations of teachers’ beliefs with their own and student outcomes. One possibility is the Job-DemandsResources Model (Bakker & Demerouti, 2007), which incorporates both positive and negative processes underlying the development of occupational well-being and can accommodate diverse

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job demands and resources, including personal resources (Bakker & Demourti, 2014). Future research may profitably adopt this framework or seek to integrate it with social cognitive perspectives to theoretically examine other teacher-related factors that are associated with their own outcomes and student outcomes. Yet another limitation concerns the limited set of teacher and student-level outcomes examined. Although the model proposed multiple theoreticallyrelevant variables at both teacher and student levels, many other factors in the classroom ecology may be expected to be impacted by teacher self-efficacy beliefs, including observed instructional and classroom management practices, work commitment, burnout, student satisfaction, and engagement (Zee & Koomen, 2016). However, the addition of factors to the model needs to be weighed against increasing the complexity of already complex latent variable multilevel models, which may pose estimation difficulties. Further limitations concern the modeling and measurement of constructs. Although interaction quality is purported to contain multiple dimensions, we represented the construct as a single factor given sizeable correlations between the constituent emotional, instructional, and organizational dimensions, which precluded examination of the way in which specific dimensions are uniquely linked to the teacher and student outcomes. Relatedly, perceptions of teacher-student interaction quality were indexed by students’ self-reports and aggregated at L2 to form an L2 aggregate of quality at the teacher level. However, it is entirely possible to index quality of teacher-student interactions at a teacher level using more “objective” measurements based on observational protocol (Pianta & Hamre, 2009). The measure of work satisfaction used in the study is another limitation. The TIMSS assessment included a measure of teachers’ satisfaction for their work generally. This stands in contrast to the math context of the study, thereby generating a lack of domain correspondence between our measures of teacher self-

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efficacy and satisfaction. Future research may profitably reexamine this model with a measure of satisfaction for teaching math to ensure optimal domain correspondence between measures of self-efficacy and outcomes in line with social cognitive theory. A final set of limitations concern the sample used in the study. First, focusing on primary school may be viewed as a limit insofar as primary teachers teach multiple subjects to their students, thereby rendering it difficult to isolate the effects of math teaching self-efficacy on math-related outcomes from beliefs about teaching other subjects. Nevertheless, per the domaincorrespondence tenet of social cognitive theory, effects of efficacy beliefs for teaching in other domains should be small or trivial as compared with math teaching efficacy on domain-specific outcomes (e.g., math achievement, interaction quality in math lessons. Furthermore, although the large samples of students and teachers is a strength, the study was limited to fourth grade Australian students and their teachers in the domain of math. Future research would do well to examine the extent to which the present findings, particularly those concerning the relations of teacher self-efficacy beliefs with student outcomes, replicate across (a) students from different grades, (b) students and teachers from different countries and educational systems, and (c) distinct domains of learning, such as English or science. Conclusions In summary, this study investigated the associations of teachers’ math teaching selfefficacy with student achievement and job satisfaction both directly and indirectly via class levels of interaction quality. Additionally, per the ecology of the classroom environment, we accounted for traditional socio-demographic, relational, and motivational predictors of math achievement at the individual level, including perceived student-teacher interaction quality and math self-concept. Although our study yielded support for the main propositions that teachers’

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self-efficacy beliefs play a role in classroom interaction quality, students’ achievement, and their job satisfaction, further developing knowledge in this area is required, including extensions to (a) longitudinal data that allow for control of prior variance in outcomes, (b) students at other grade levels, and (c) other academic domains (e.g., English, science).

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Table 1. Fit statistics and Goodness-of-Fit Indices for the Multilevel CFA and SEM models. χ2

df

CFI

TLI

RMSEA

Multilevel CFA

3270.346*

523

.931

.922

.029

Multilevel CFA

3264.147*

532

.932

.923

.029

3329.819*

534

.930

.922

.029

Model

with invariance loadings Multilevel SEM

Note. df = degrees of freedom; CFI = comparative fit indices; TLI = Tucker-Lewis index; RMSEA = root mean square error of approximation. * p < .001

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Table 2. Correlations (and standard errors) among the L1 Factors and Manifest Variables. 1 1. Student gender

2

3

4

5

–

2. SES

.015 (.022)

–

3. Interaction Quality

.036 (.016)*

.006 (.023)

–

4. Math self-concept

-.170 (.015)***

.169 (.021)***

.286 (.018)***

–

5. Math achievement

-.078 (.015)***

.303 (.021)***

.068 (.018)***

.549 (.013)***

–

Note. Nstudents = 6057; Nteachers = 452; Student gender: 0 = male; 1 = female; SES = socio-economic status. * p < .05, ** p < .01, p < .001.

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\Table 3. Correlations (and standard errors) among the L2 Factors and Manifest Variables. 1 1. Teacher gender 2. Years’ Experience

2

3

4

5

– .157 (.045)**

–

-.069 (.048)

.112 (.050)*

–

4. Interaction Quality

.225 (.071)**

-.153 (.066)*

.111 (.063) +

–

5. Math achievement

.067 (.063)

.095 (.053)+

.159 (.058)**

.115 (.083)

–

6. Work satisfaction

.037 (.052)

-.082 (.047)+

.244***

.167 (.069)*

.137 (.062)*

3. Teacher Self-Efficacy

6

(.060)*** Note. Nstudents = 6057; Nteachers = 452; Teacher gender: 0 = male; 1 = female. + p < .10, * p < .05, ** p < .01, p < .001.

54

–

Table 4. Slope coefficient estimates from the retained multilevel SEM model. Effect

UStd. (SE)

Std. (SE)

Level 2 Teacher self-efficacy  job satisfaction

0.285 (0.071)***

.246 (.062)***

Teacher self-efficacy  interaction quality

0.035 (0.014)*

.152 (.061)*

Teacher self-efficacy  math achievement

0.101 (0.045)*

.143 (.062)*

Interaction quality  math achievement

0.262 (0.277)

.085 (.089)

Interaction quality  job satisfaction

0.572(0.399)

.114 (.075)

YrsExp  interaction quality

-0.003 (0.001)**

-0.019 (.006)**

YrsExp  job satisfaction

-0.007 (0.004)*

-0.009 (0.004)*

YrsExp  teacher self-efficacy

0.008 (0.003)*

0.011 (0.004)*

YrsExp  math achievement

0.004 (0.003)

0.007 (0.005)

Gender  interaction quality

0.118 (0.033)***

0.691 (.182)***

0.096 (0.128)

0.112 (.149)

-0.169 (0.094)+

-0.227 (.125)+

0.058 (0.092)

0.110 (0.175)

Gender  job satisfaction Gender  teacher self-efficacy Gender  math achievement

55

Level 1 Math self-concept  math achievement

0.931 (0.033)***

.566 (.013)***

Interaction quality  math achievement

-0.177 (0.033)***

-.095 (.017)***

Interaction quality  math self-concept

0.334 (0.027)***

.293 (.018)***

Gender  math achievement

0.030 (0.022)

0.036 (0.027)

Gender  math self-concept

-0.184 (0.016)***

-0.361 (0.029)***

Gender  Interaction quality

0.032 (0.015)*

0.072 (0.033)*

0.124 (0.011)***

.233 (0.021)***

SES  math achievement

Note. UStd = unstandardized estimate; Std = standardized estimate; SE = standard error. Teacher gender is coded as a binary variable with 0 = male and 1 = female; Student gender is coded as a binary variable with 0 = male and 1 = female. Effects of binary covariates (i.e., gender) and years of teaching experience on the latent outcomes are partially standardized coefficients using only the variances of the latent variables for standardization.

56

Figure 1. Heuristic model of teacher self-efficacy in relation to classroom processes, student outcomes, and teacher outcomes. Adapted and modified from Zee, M., & Koomen, H. M. Y. (2016). Teacher self-efficacy and its effects on classroom processes, student academic adjustment, and teacher well-being: A synthesis of 40 years of research. Review of Educational Research, 86, 9811015. doi: 10.3102/0034654315626801

57

Figure 2. The multilevel SEM model tested in the current study. Note. Covariance terms are not included to avoid clutter. Observed indicators are stylistic to avoid cluster with the precise number of indicators included in parentheses. Covariate relations are shown with dashed arrows. Gender (S) = student gender; Gender (T) = teacher gender; SES = socio-economic status. 58

Highlights 

An integrative model positing relations of math teaching self-efficacy with teacher and student outcomes was examined.



Additionally, traditional student level relational and motivational predictors of achievement were integrated into the model.



Math teaching self-efficacy was positively associated with class-average math achievement and student-teacher interaction quality.



Math teaching self-efficacy was positively associated with teachers’ job satisfaction.

59