The role of teacher characteristics for student achievement in mathematics and student perceptions of instructional quality

ABSTRACT This study explores how aspects of teacher quality are related to student achievement and students’ perceptions of instructional quality on the basis of eighth grade TIMSS 2011 data for Sweden. Indicators of teacher quality are coursework in mathematics as measured by the number of semesters of studying mathematics, years of teaching experience and teacher self-efficacy beliefs. The study employs confirmatory factor analysis and structural equation modelling as primary methods. Results indicate that teachers with higher self-efficacy beliefs were rated by students as delivering higher instructional quality. However, this was not reflected in student achievement levels. Instead, with student socio-economic and immigrant background under control, there was a significant positive relationship between coursework in mathematics and student mathematic achievement levels, as well as between student perceptions of instructional quality and achievement. Relations between teaching experience and student achievement followed a non-linear pattern, with the effect of teaching experience increasing up to 19 years and declining afterwards.


KEYWORDS
Teacher quality; instructional quality; student achievement; mathematics; Trends in Mathematics and Science Study (TIMSS) Background During recent decades research has been able to show that the classroom level is more important for student results than the school level, and that teachers vary systematically in effectiveness (Muijs et al., 2014). Despite the convincing evidence that teacher quality is crucial for student success, there is yet no agreement on which teacher characteristics are of the greatest importance for student learning outcomes (Darling-Hammond, 2006;Gustafsson, 2003;Hattie, 2009;Rivkin, Hanushek, & Kain, 2005;Scheerens & Blömeke, 2016). Traditionally in educational research, teacher quality has been measured by formal qualifications and experience. Numerous studies have investigated the effects of teacher educational background, level of education, certification status and years of teaching experience on student achievement (Darling-Hammond, 2000Goe, 2007;Wayne & Youngs, 2003). Other researchers have considered teacher personal characteristics, such as selfefficacy, locus of control or verbal intelligence to be crucial for teacher success (Scheerens, 2016).
Consequently, there appear to be a number of teacher characteristics that may contribute to explaining variation in teacher effectiveness. While many studies have explored a single aspect of teacher quality in relation to student outcomes, or several indicators of formal competence only (e.g. teacher education level and certification status), fewer have combined indicators of formal qualifications with perceived competence, e.g. teacher self-efficacy beliefs. However, according to Scheerens and Blömeke (2016), teacher quality is a multi-faceted construct, comprising both cognitive (knowledge) and non-cognitive aspects (beliefs, attitudes etc.). Therefore, relating indicators from both aspects of teacher quality to student outcomes may provide a more precise explanation for variation in teacher effectiveness (Goe, 2007;Cochran-Smith & Zeichner, 2005).
In this study, we investigate relations between cognitive and affective teacher characteristics, student mathematics achievement, and student perceptions of instructional quality.

Theoretical and conceptual framework
This study is placed within teacher effectiveness research strand, which emerged in the early 2000s within a broader framework of educational effectiveness, and which focuses on teaching as facilitated learning through learning activities and student engagement (Scheerens, 2005). The basic model of educational effectiveness aimed to open the "black box" of a classroom, revealing teacher factors which contribute to student cognitive and non-cognitive outcomes (Scheerens, 2016). It was further developed into the dynamic model, which not only identified factors contributing to variation in student outcomes, but also attempted to explain the interrelation between these factors at different levels through the application of relevant theories (Creemers & Kyriakides, 2006). Teacher characteristics (comprising professional knowledge, motivation and beliefs) are considered as model inputs, yet are recognised to be amenable through teacher education, experience and professional development (Scheerens, 2005).
A central construct underpinning the analyses in the present study is the notion of teacher knowledge, which has been classified into content knowledge and pedagogical content knowledge in the seminal work by Shulman (1986). Content knowledge (CK) refers not only to the mere knowledge of facts and concepts in a domain, but also to its substantive and syntactic properties, i.e. "the teacher need not only understand that something is so, the teacher must further understand why it is so…" (Shulman, 1986, p. 9). Pedagogical content knowledge (PCK), in turn, covers the subject-matter knowledge for teaching, or the means of presenting the subject in a way that is comprehensible to students.
In recent decades researchers have attempted to revisit Shulman's conceptualisations of CK and PCK. One of the most influential re-conceptualisations is mathematical knowledge for teaching, or MKT (Hill, Ball, & Schilling, 2008). MKT is a global construct comprising subject-matter knowledge and pedagogical content knowledge. In contrast, other researchers have claimed that CK and PCK are distinct, yet related constructs (Kleickmann et al., 2013). Additionally, CK has been found to be a prerequisite for PCK development, i.e. teachers need profound subject matter knowledge for making content comprehensible for students (Friedrichsen et al., 2009;Krauss et al., 2008). While a variety of definitions of content and pedagogical knowledge have been suggested, throughout this study we draw on Shulman's original conceptualisations, and treat CK and PCK as two distinct constructs. Teacher knowledge comes from various sources, such as teachers' own educational experiences, teacher education, and teaching experience (Friedrichsen et al., 2009). Regardless of the quality of knowledge, putting it into practice may be challenging without positive self-efficacy beliefs.
Social cognitive theory (Bandura, 1997(Bandura, , 1986 underscores the role of human agency within a triadic interaction of individual, behavioural and environmental factors. The idea that individuals are able to influence their actions has laid the foundations for the development of the self-efficacy construct. In a socio-cognitive perspective, self-efficacy is defined as "people's judgements of their capabilities to organise and execute courses of action required to attain designated types of performance" (Bandura, 1986, p. 391). Applied to the teaching profession, self-efficacy beliefs shape the teacher's effort and persistence, resilience towards failures, and the stress level in response to demanding tasks. Teacher self-efficacy is composed of the key domains of teachers' work: instructional, motivational and classroom management self-efficacy (Tschannen-Moran, Woolfolk Hoy, & Hoy, 1998).
According to Bandura (1997), performance accomplishments, or prior teaching experience, is a major source of teacher self-efficacy. A key characteristic of teacher self-efficacy is that it is a task, domain, and context-specific construct (Bandura, 1997(Bandura, , 2006. As such, teachers' self-efficacy depends on the subject-matter taught, as well as on the students being taught (Tschannen-Moran et al., 1998).
Teachers' perceived competence may not necessarily correspond to the actual quality of instruction. Teacher instructional quality can be conceptualised as instructional behaviour able to predict and/or explain student educational outcomes (Weinert, Schrader, & Helmke, 1989). It has been described as comprising the following key aspects: classroom management, instructional clarity, cognitive activation and supportive climate (Klieme, Pauli, & Reusser, 2009). In addition, an aspect of instructional quality aimed at student engagement has also been identified (Scherer & Gustafsson, 2015).
Thus, teachers possess certain levels of professional knowledge, of which important aspects are CK and PCK, that is, knowledge of content and knowledge of how to effectively teach this content to students from various backgrounds. Teachers also have certain levels of confidence in teaching their subject. Theoretically, such teacher characteristics should be related to student perceptions of instructional quality and student achievement. A more detailed account of previous findings on the relationships between teacher characteristics, student achievement, and student perceptions of instructional quality, is described in the following section.

Previous research
In this section, earlier research on the relations between teacher education, teaching experience, teacher self-efficacy beliefs, instructional quality and student outcomes is presented. The need to consider student background factors is addressed in a brief account of the Swedish school context.
Teacher education and teaching experience Educational research commonly employs teacher educational level, certification status and coursework in the subject as indicators of teacher knowledge. Wayne and Youngs (2003) emphasise that measures of teacher knowledge are subject-and grade-specific. Domain specificity has been advanced as a key factor for successful teaching. In their comprehensive meta-analysis, Seidel and Shavelson (2007) concluded that teachers need to have an in-depth understanding of the content and the nature of the domain that is pedagogically useful. In a review, Goe (2007) presented inconsistent findings on the relative contributions of teacher qualifications such as teachers' coursework, level of education, certification, test scores and experience on student achievement across subjects and grades. However, in case of mathematics, it was concluded that teacher qualifications do matter, in particular at the secondary school level. Domain and gradespecific results of subsequent studies have corroborated these conclusions (e.g. Baumert et al., 2010).
Still, there is some disagreement on which qualifications make substantial contributions to teacher effectiveness. This may be in part due to differences between teacher education programs across and even within countries (Blömeke, Olsen, & Suhl, 2016;Wayne & Youngs, 2003). Further, some researchers within education production function research have contended that neither formal teacher qualifications, nor teaching experience account for variation in teacher effectiveness (Hanushek & Rivkin, 2012). Interestingly, researchers within the same research strand have found effects of teaching experience on student achievement, albeit only in the early career years.
Indeed, research on effects of teaching experience on student achievement is generally inconsistent Rice, 2010). Moreover, interpretations of experience effects (or a lack thereof) is challenging for a number of reasons (Rice, 2010;Wayne & Youngs, 2003). First, studies on the effects of teaching experience employ different methodologies (Kukla-Acevedo, 2009). Secondly, the effects of teaching experience may capture the influence of labour market conditions for teachers (Wayne & Youngs, 2003). Finally, similar to teacher education, the effects of teaching experience are likely to vary across subject domains and grade levels (Goe, 2007). Goe's synthesis provides evidence that teaching experience matters particularly for student mathematics achievement, a proposition which has found support in subsequent studies (e.g. Wiswall, 2013). However, a common feature of the studies investigating teaching experience is that the association between the two is often non-linear, and the effects of teaching experience are more pronounced for teachers with less than five years in service (Rice, 2010).
Thus, researchers within the education production function tradition have supported the claim that the first few years of teaching exert the greatest influence on student achievement (Clotfelter, Ladd, & Vigdor, 2007;Hanushek & Rivkin, 2012). Further, a number of studies have suggested that teachers reach a certain peak in their careers, after which their effectiveness declines. For example, Chingos and Peterson (2011) suggested a downslide in the effect of mathematics teachers' experience after approximately 15 to 20 years. As summarised by Rice (2010), experience matters, but additional experience does not always enhance student achievement.
Adding to the controversy regarding the contribution of experience to teacher effectiveness is the argument that teacher education programs may be equivalent to the effects of early career experience (Darling-Hammond, 2000). As Darling-Hammond (2014) explained, longer periods of teaching experience within teacher education programs allow for possibilities to apply theoretical knowledge during classroom teaching practice. This combination of theory and practice can mean that recently graduated teacher education students are as effective as their more experienced colleagues. This argument is further corroborated by a cross-sectional study of German preand in-service mathematics teachers (Kleickmann et al., 2013), where teacher education was found to be more important for the development of CK and PCK than teaching experience and formal professional development.
However, as argued by Kukla-Acevedo (2009), measures of teacher content knowledge, such as number of courses taken and the nature and level of the degree, might not fully capture teacher skills in teaching the subject in the classroom, i.e. PCK. Thus, teaching experience could be considered an important source of PCK, and it certainly plays a vital role in bridging CK and PCK (Friedrichsen et al., 2009). Moreover, teaching experience serves as a key source of self-efficacy beliefs (Bandura, 1997).

Teacher self-efficacy beliefs
Teacher self-efficacy beliefs are considered to be an important aspect of teacher competence, influencing teachers' instructional behaviours and student motivation (Klassen & Tze, 2014;Klassen, Tze, Betts, & Gordon, 2011;Tschannen-Moran & Woolfolk Hoy, 2001;Tschannen-Moran et al., 1998). At the same time, teacher self-efficacy research has suffered from theoretical and conceptual confusion (Henson, 2002). Therefore, a number of researchers have called for more evidence on the links between teacher self-efficacy and student learning outcomes at the classroom level (Wheatley, 2005;Wyatt, 2014;Zee & Koomen, 2016).
It is worth noting that in studies following Bandura's theoretical guidelines for the conceptualisation and measurement of teacher self-efficacy, only a modest association of teacher self-efficacy with student achievement was found (Henson, 2002;Klassen et al., 2011;Wyatt, 2014). A further scrutiny of these studies reveals a lack of domain-and grade-specific instruments, thus questioning the assumption of a link between the two (Zee & Koomen, 2016).
Constructs related to teacher instructional practices have revealed more consistent relations with teacher self-efficacy, in particular at the primary school level (Tschannen-Moran et al., 1998;Zee & Koomen, 2016). For example, Thoonen, Sleegers, Oort, Peetsma, and Geijsel (2011) concluded that teacher self-efficacy was an important factor enhancing primary school teachers' instructional quality.

Instructional quality
The transformation of teacher beliefs into practice is a complex process (Fives & Buehl, 2012). Moreover, a number of researchers have stressed the importance of exploring teacher knowledge and skills together with teacher beliefs. For example, Raudenbush, Rowan, and Cheong (1992) contend that self-efficacy is an important, but not sufficient, factor for successful teaching, as teachers may lack the necessary knowledge and skills to enact their beliefs. In addition, in order to understand how teacher knowledge and beliefs are connected to student learning, we might first need to understand how they are translated into teacher behaviour, and lead to instructional quality (Fives & Buehl, 2012).
Instructional quality is traditionally assessed by either students, teachers or external observers (Wagner, Göllner, Helmke, Trautwein, & Lüdtke, 2013). Student-assessed instructional quality has been employed in multiple ways in educational research. For example, Wagner et al. (2016) have used it as a predictor of student mathematics achievement. It has also served as an outcome of mathematics teachers' self-efficacy beliefs (Holzberger, Philipp, & Kunter, 2013). Thus, in a longitudinal study of the reciprocal effects between mathematics teachers' self-efficacy and ninth grade student ratings of instructional quality, Holzberger, Philipp & Kunter were able to confirm the effect of student-assessed instructional quality on teacher self-efficacy. However, a causal effect of teacher self-efficacy on the quality of instruction was only partially established. Finally, instructional quality has been explored as a mediating variable between teacher knowledge and student outcomes . Similar to indicators of teacher competence, instructional quality should be studied in relation to student grade and subject matter taught (Scherer & Gustafsson, 2015). Besides dependence on contextual factors, instructional quality is also contingent on the national educational setting, such as the structure of educational system, the national curriculum, and the organisation of teacher education (Weinert et al., 1989).

Addressing student background in a national context
It is well-established that student socio-economic status (SES) largely affects student achievement (Yang Hansen & Gustafsson, 2016). Previous research has found, in most countries, the influence of individual student background on achievement to be quite stable over time (Gustafsson, 1998). However, recent research indicates that in the case of Sweden, the introduction of a market-based school system in the early 1990s has resulted in increased segregation between schools with respect to student socio-economic background (Vlachos, 2011). Additionally, the school SES-effect has strengthened over time (Yang Hansen, Rosén, & Gustafsson, 2011).
A large variety of indicators of SES have been used in educational research (e.g. White, 1982). According to Sirin (2005), conceptualisation of SES should consider (a) the unit of analysis for SES-data (findings are likely to be contaminated when aggregated data are used to make assumptions at the student level), (b) the type of SES measure (it affects relationships with student outcomes), (c) the range of the SES-variable (dichotomous variables are less likely to produce strong correlations), and (d) the source of SES-data (accuracy of reports is dependent on student family background, age and achievement level).
The number of books in a student's home is commonly used in large-scale assessments of student achievement as a proxy for student SES (Mullis, Martin, Ruddock, O'Sullivan, & Preuschof, 2009). Even though it has long been considered a useful measure of student background (Schütz, Ursprung & Woessmann, 2008), its validity has been questioned. For example, Engzell (2016) recommended caution in reliance on the number of books at home as a sole indicator of socio-economic background, as underreporting by lowachievers and endogeneity of parental input may contribute to upward bias. Nevertheless, this indicator has been well tested in previous international research for many decades (Hanushek & Woessman, 2011), and in combination with a measure of the language spoken at student's home, it can be considered satisfactory to serve its purpose. Student academic achievement is largely dependent on proficiency in the language of school instruction (OECD, 2015). Similar to student socio-economic background, it is a proxy for immigrant background on the individual level, while on the aggregated level, it is indicative of classroom composition. Sirin (2005) found SES to have less predictive power for minority students. Previous research has found that while immigrant background is generally associated with lower school achievement levels, in many countries this relation may be partly attributed to an interaction with student SES, with immigrant children more often lacking family resources favourable for successful studies (Preevo, Malda, Mesman & van IJzendoorn, 2016). On the other hand, Hansson and Gustafsson (2013) analyzed TIMSS data for eighth graders in Sweden and did not find SES to have a reduced relationship with mathematics achievement for groups of students with a foreign background. However, in their study SES was measured by several indicators of family educational resources, thus covering a multi-dimensional construct.
In case of the current study, where interest is directed to relationships between teacher characteristics and student outcomes, it is worthwhile exploring these relations for students with different background characteristics. As previously mentioned, student SES and immigrant background both bear disparate meanings at the individual and class level, and this can be addressed by employing two-level analysis.

Teacher situation in Sweden
Swedish educational reforms of the 1990s concerned decentralisation, school choice and a voucher system. The expansion of the independent/private school sector was intended to enhance quality through increased competition between schools (Yang Hansen & Gustafsson, 2016). In addition, teacher education in Sweden was altered several times since the 1990s, with varying emphasis on the importance of subject and grade specialisation for teachers (Hansson & Gustafsson, 2016). Besides, teacher quality is not evenly distributed between schools, with the number of certified teachers in public compulsory schools being 80%, compared to 67% in independent schools (The Swedish National Agency for Education, 2017).

Summary of previous research
Summarising previous research on teacher knowledge and beliefs, and their effects on student outcomes, it can be concluded that while there is a plethora of studies conducted in a North American setting, in Europe most studies have been conducted in Germany. In particular, German researchers have addressed the subject domain and grade level relevant for the current study, i.e. secondary-level mathematics (Baumert et al., 2010). In the Nordic region, relations between teacher self-efficacy beliefs, teacher education and instructional quality have been primarily investigated in the Norwegian context (e.g. Nilsen & Gustafsson, 2016;Skaalvik & Skaalvik, 2007).
In Sweden, research on the indicators of teacher competence has been more sporadic. For example investigations into teacher formal competence (Frank, 2009;Myrberg, 2007), self-efficacy beliefs (Lander, 2013), and instructional responsibility (Hansson, 2010) have been carried out. When the effects of teacher competence on student achievement have been studied, it is mainly in relation to primary school students' reading achievement (Myrberg, Johansson, & Rosén, 2018). Research combining both cognitive and affective teacher characteristics, and dealing specifically with mathematics teachers at the secondary school level in the Swedish setting is lacking.

Aims
In the light of this background, the study aims to address the following research questions: (1) How are the number of semesters teachers studied mathematics, years of teaching experience and self-efficacy beliefs related to 8th grade student mathematics achievement and student perceptions of the quality of instruction in mathematics? (2) Do relationships between the three indicators of teacher quality and student outcomes, as measured by student mathematics achievement levels and student perceptions of the quality of instruction in mathematics, vary depending on classroom composition with respect to student socio-economic and immigrant background? (3) Are student achievement levels and perceptions of instructional quality within classrooms related to individual student socio-economic and immigrant background?

Data sources and sample
This study uses data from Swedish participation in TIMSS 2011 (Trends in International Mathematics and Science Study). TIMSS is one of the studies of the International Association for the Evaluation of Educational Achievement (IEA), where mathematics and science achievement of fourth and eighth grade students is assessed. TIMSS collects educational achievement data and information about the educational contexts for learning mathematics and science (Mullis et al., 2009). The study is conducted on a four-year cycle, with the first assessment held in 1995, and with a forthcoming study in 2019. In Sweden, a total of 153 schools, 296 teachers and 5573 students in grade 8 participated in TIMSS 2011 (The Swedish National Agency for Education, 2012).

Variables and measures
Student achievement scores in mathematics, teacher and student background variables, as well as variables indicating teacher self-efficacy and student-assessed instructional quality were retrieved from the international TIMSS database (see Table 1). TIMSS provides a standardised mathematics achievement score for each student. Given a large content coverage, the study employs a complex matrix-sampling booklet design, where each student only answers a part of the total item pool (Foy, Aurora & Stanco, 2013). In order to obtain accurate student and classroom-level results on the whole assessment, the plausible value methodology is used. International assessments like TIMSS use five plausible values, which are latent, i.e. not directly observable (Wu,  .73 (i) -TIMSS international questionnaire item, (n) -item in the national extension of the international questionnaire 2005). The variability between these latent variables encapsulates the uncertainty inherent in the scale estimation process. Because five plausible values are provided, analyses involving achievement score are run separately for each PV, and the results of the five analyses are combined into a single value by averaging the resulting statistics (Foy, Brossman, & Galia, 2011).
In the first TIMSS study in 1995, the international mean of the scale was set to 500 points, with a standard deviation of 100 (Mullis, Martin, Foy, & Arora, 2012). The data from successive studies are transformed to the same metric to ensure trend comparability over time (Foy et al., 2011).

Student background variables
In the current study, student background variables are used at both individual (within) and classroom (between) levels. Thus, student reports of "The number of books at home" is used as a proxy for student socio-economic background at the individual level, and as an indicator of classroom socio-economic composition at the classroom level. Similarly, the variable "Language spoken at home" serves as a proxy for student immigrant background at the individual level, and as an indicator of classroom immigrant composition at the classroom level.

Teacher background variables
For teacher content knowledge (CK), "The number of semesters of studying mathematics" is used as a proxy. For pedagogical content knowledge (PCK), "Years of teaching experience" is used as a proxy. It should be noted that there were other teacher characteristics in the TIMSS teacher background questionnaire, which turned out to be of a limited utility for the current study due to e.g. lack of variation in teacher responses (most teachers indicated having the highest educational level, i.e. University). Two latent variables were formulated in the current study, teacher self-efficacy (T_SE) and student-assessed instructional quality (INQ).
Teacher self-efficacy TIMSS assessed teacher self-efficacy in teaching mathematics through items in the "Confidence in teaching mathematics scale", which is based on Bandura's theoretical foundation for the self-efficacy construct (Mullis et al., 2009). In addition, the international scale was complemented by the national self-efficacy items in the Swedish extension, which were taken advantage of in this study.
Among the three theoretical dimensions of teacher self-efficacy identified by Tschannen-Moran and Woolfolk Hoy (2001)instructional, motivational and classroom managementonly instructional and motivational dimensions were covered by the TIMSS items. Instructional self-efficacy was reflected in the items regarding teachers' confidence "to adapt teaching to meet individual students' needs", "to apply a variety of teaching strategies" and "to help underachieving students". Motivational self-efficacy was measured by the items about teachers' confidence "to adjust teaching to wake students' interest in mathematics", "to help students understand value of learning mathematics" and "to motivate students with low interest for mathematics".
Responses were given on a 3-point scale ranging from "Very confident" to "Not confident". Cronbach's reliability index for this six-item scale was .74. Descriptive statistics for the items in teacher self-efficacy (T_SE) scale are presented in Table 1. Mathematics teachers in the Swedish sample report above average levels of self-efficacy in all aspects. Mean self-efficacy levels appear to be nearly the same across all dimensions, with the exception of the somewhat lower mean for teacher self-efficacy to motivate students with a low interest in mathematics.

Instructional quality as assessed by students
The TIMSS student background questionnaire included several aspects of instructional quality that were used to form a latent variable of instructional quality (INQ): student assessments of instructional clarity ("I know what my teacher expects of me", "My teacher is easy to understand"), classroom management ("There is a good working environment during mathematics lessons"), supportive climate ("My teacher helps when I have difficulties with mathematics") and student engagement ("My teacher gives me interesting tasks").
Responses were given on a 4-point scale, ranging from "Agree a lot" to "Disagree a lot". Cronbach's reliability index for this five-item scale was .73. Student responses to these items were recoded so that lower values reflect lower assessment of instructional quality. Students reported high levels of teacher's help with difficulties with mathematics, but rather low levels of interest in mathematics tasks given by the teacher. They also rated the quality of the classroom working environment relatively low.
Along with the student achievement variable (M_ACH), INQ was used as an outcome variable in the study.

Two-level confirmatory factor analysis (CFA) with Structural Equation Modeling
(SEM) were used as main methods. These analytical approaches allow the formulation of measurement models where a set of observable indicators are used as measures of one or more latent variables, which can be subsequently related in structural models (Brown, 2015). A latent variable defines a construct which is not directly observable through single indicators, and is useful when theoretical constructs, like self-efficacy, are operationalised (Bollen, 2002). Another strong feature of latent variables is that they are devoid of measurement error, since the unique part of the variance is separated from the unexplained part (Gustafsson, 2009).
In addition, educational assessment data, such as in the TIMSS study, often have a hierarchical observational structure, with students being nested in classrooms. Such hierarchically structured data are possible to analyze with the help of multilevel SEM (Gustafsson, 2009). Since individuals belonging to the same cluster (same classroom) tend to be more alike than two randomly selected individuals, two-level modelling is used to account for dependencies between individual observations. An assumption of independence would render the standard errors produced by conventional statistical tests too small, which could lead to spuriously significant results (Hox, 2002). Thus, although the focus of the study is to investigate teacher effects at the classroom level, student level variables were included in the analysis.
Weights for each hierarchical level, student and classroom, were used (Asparouhov, 2006). Since the focus of the analysis is the classroom level, weights at the classroom level are a product of a class weighting factor, a class weighting adjustment, a school weighting factor, and a school weighting adjustment. At the student level, weights are a product of a student weighting factor and a student weighting adjustment (Foy et al., 2013).
The analyses were conducted using the computer program M-plus 7. Model fit was evaluated using the recommended fit indices, Chi2, RMSEA, CFI and SRMR (Brown, 2015). There are different recommendations regarding which fit indices and cut-off values should be used to assess model fit. For RMSEA (root mean square error of approximation), we use the recommended values of 0.07 or below for an acceptable fit (Hu & Bentler, 1999). For CFI (comparative fit index), which is a fit index that depends on the average size of the correlations in the data, values of 0.95 and above are considered as acceptable, while values close to 1.0 are preferred (ibid., 1999). For SRMR (standardised root mean residual), which is a measure of residuals computed separately for within and between level, values below 0.08 are deemed acceptable (Hu & Bentler, 1999). In addition, for latent measurement models, the factor loadings of .30 and higher render respective indicators reasonable measures of their latent construct (Brown, 2015).
The proportion of missing data was about 15-20% in the teacher background questionnaire. Most of this pertains to the complete questionnaire rather than single items. In a missing data analysis, no systematic missing values were discovered. In the student questionnaire the proportion of missing data was only minor, about 2-3%.
In the first step, latent measurement models were formulated using confirmatory factor analysis, and in a second step, relationships between variables were specified through structural equation modelling. A latent measurement model of student-assessed instructional quality was formulated at both individual student and classroom level. As the focus of our study was on relations between teacher characteristics and student outcomes, we were primarily interested in the aggregated level of student perceptions of instructional quality. Students were considered as informants of their shared learning environment, and classroom level analysis is warranted when investigating such a group-level construct (Lüdtke, Robitzsch, Trautwein, & Kunter, 2009). In addition, it is important to examine psychometric properties of such reflective constructs, by means of e.g. level-two reliability test (Lüdtke et al., 2008). A latent approach adopted in this study takes unreliability of the group mean into account.
The variables 'books at home' (BOOKS) and "language spoken at home" (LANG) indicating student socio-economic and immigrant background were used as controls at the individual student (within) and classroom (between) levels. Zero-order correlations for manifest variables at the classroom level are presented in Table 2.

Results
In this section, latent measurement models are formulated and evaluated, and results of structural models are presented.  First, a measurement model of teacher self-efficacy was tested. Factor loadings for teacher self-efficacy were quite evenly distributed among the six indicators, with loadings ranging from .55 to .77. It is reasonable to believe that the items "Teacher self-efficacy in helping underachieving students" and "Teacher self-efficacy in adapting teaching to meet individual student needs" have substantive commonalities, since both relate to providing assistance to struggling students. The modification index for these indicators pointed to the correlated uniqueness of the residuals. Thus, in the re-specified model, a residual covariance was included. Factor loadings and model fit for the T_SE variable are presented in Table 3.

A latent model of teacher instructional quality
Second, we tested a measurement model of teacher instructional quality. Since this construct was assessed by students, we formulated the measurement model of teacher instructional quality at both the within and the between-classroom levels. To conduct a two-level analysis, it was necessary to ascertain whether the between-class effects were present for the instructional quality variables. Therefore, intraclass correlations were examined. When the intraclass correlations are .05 or higher, two-level modelling is warranted (Muthén, 1994). In the present case, the intraclass correlations were about .05-.15 for the INQ variables. Factor loadings at two levels along with fit indices for the whole model are exhibited in Table 4. The model fitted the data well.
No significant association between the three teacher characteristics and student achievement was present. Teacher self-efficacy beliefs were positively linked with teaching experience (.23) and the number of semesters of studying mathematics (.21). A summary of the model estimates and model fit (Models 1 to 3) are shown in Table 5.
Relations between teacher characteristics and student-assessed instructional quality Further, a structural model with the above teacher characteristics and student-assessed instructional quality as a dependent variable was constructed. The number of semesters of studying mathematics and teaching experience were not associated with student aggregated perceptions of instructional quality. Teacher self-efficacy was significantly positively related to student-assessed instructional quality at .23, meaning that teachers  Table 6.

Considering student classroom composition
In a further step, and in order to control for student classroom composition, proxies for student SES (BOOKS) and student immigrant background (LANG) were introduced in the model as explanatory variables stepwise. The effect of BOOKS on student achievement was .85, and increased to .88 after controlling for LANG (Models 4-5 in Table 5). There was no significant association between LANG and M_ACH, however BOOKS and LANG were highly correlated (−.49). Additionally, no association was found between T_SE and BOOKS, pointing towards independence of teacher self-efficacy beliefs from student socio-economic classroom composition. However, a positive correlation (.21) of T_SE with LANG was present, indicating that teachers felt more efficacious teaching students with an immigrant background.  After controlling for BOOKS, a positive significant relationship between T_MATH and M_ACH (.16) became apparent. This effect slightly increased (.17) after LANG was introduced into the model. Teachers who had studied more semesters of mathematics thus tended to teach students in lower-SES classes. A cross-tabulation of BOOKS and T_MATH further confirmed this result. In order to investigate possible interaction effects (i.e. the presence of a higher effect of T_MATH on achievement in high or low-SES classrooms), an interaction term was introduced between T_MATH and BOOKS using aggregated data. However, no significant interaction was estimated.
There was no association between BOOKS or LANG with student-assessed instructional quality (INQ) at the classroom level (Model 5 in Table 6). However, INQ was positively associated with M_ACH at .20, meaning that better achieving students tended to give higher ratings of the instructional quality provided by their teachers.

Differences in achievement and ratings of instructional quality for students with differing levels of SES and immigrant background on individual level
At the individual level, BOOKS was positively related to M_ACH at .27, meaning that students from more advantageous backgrounds obtained higher scores on the mathematics test. LANG in turn, had a minor negative association with mathematics achievement of −.05, implying that immigrant students scored somewhat lower on the mathematics test than did Swedish students (model 5 in Table 5).
BOOKS and LANG had positive, although minor, associations with INQ on the individual level. These were .04 and .08 respectively. This implies that, within classes, students from more advantageous socio-economic backgrounds, as well as students with an immigrant background, gave higher ratings to the instructional quality of their teachers with classroom composition kept under control (model 5 in Table 6).

Elucidating the role of teaching experience
Our initial results showed an insignificant association of teaching experience with student achievement. Based on previous research findings, we tested for nonlinearity between teaching experience (T_EXP) and student mathematics achievement (M_ACH). After introducing a quadratic term of teaching experience in the model, a positive linear coefficient of 1.93 and a negative quadratic coefficient of −.05 were obtained. This suggests an increase in the effect of teacher experience up until a certain point, after which it declines. The coefficients obtained allowed us to estimate this pointthe vertex of the parabolawhich was 19 years. The relation is displayed in Figure 2 below.
Thus, the association of teaching experience with student achievement was at a peak at 19 years, gradually declining afterwards. More specifically, adopting a curvilinear approach revealed that, for example, any given class taught by a teacher with 15 years of experience was associated with an average class achievement of 487 points, while classes taught by a teacher with 5 years of experience were associated with an average class achievement of 477 points. This can be equated to a difference of approximately one half of a semester of schooling (Luyten & Veldkamp, 2011).

Teacher characteristics and student achievement
Results of the study demonstrated a direct positive association between the number of semesters teachers studied mathematics, and student achievement. This finding was anticipated, and is concordant with previous research which points to the importance of teacher content knowledge for student achievement, particularly in mathematics (Baumert et al., 2010). That the effect of teacher coursework became apparent after controlling for student socio-economic background, can be attributed to the fact that lower-SES classes had teachers who had studied the subject for longer time. In the Swedish context, this result may be further explained by the uneven distribution of teacher quality, where public schools have a larger proportion of well-qualified teachers than independent schools (Myrberg & Rosén, 2006). Still, additional coursework in mathematics did not enable teachers to compensate for students' disadvantaged background, as demonstrated by the study results. As suggested by Hansson (2012), this could be due to the fact that teachers' responsibility for guiding these students' learning is limited compared to students in higher SES classes who receive greater teacher support in learning mathematics.
With regards to teaching experience, the results of this study suggest its non-linear relationship with student achievement. Our findings indicate that the effect of teaching experience on student achievement increases up to 19 years of experience. This is consistent with recent findings by Papay and Craft (2016), who challenge the widespread notion of a "performance plateau" reached by teachers after just a few years in service. Indeed, the idea that teachers cease to contribute to student learning shortly after embarking on their careers is not in agreement with research on the importance of PCK, and how it is acquired during field experience (Friedrichsen et al., 2009).
Our results are also in line with those by Chingos and Peterson (2011), who suggest the possibility of a decline in the effect of experience after 15 to 20 years of teaching. This curvilinear relationship may have many and interacting causes. Viewed as a link between teachers' content and pedagogical content knowledge, experience allows teachers to apply their theoretical knowledge into actual classroom teaching, and to further refine their teaching skills, which thus contributes to their effectiveness. This process may be particularly rapid in the beginning of the teaching career. Yet, according to our findings, teachers continue to learn on the job long past this period. The flattening of the effects of experience on student achievement after nearly two decades in service may, for example, be due to teachers' lack of opportunities to upgrade knowledge through professional development programs (Rice, 2010). Previous research has also found that teachers in the later stages of their careers can be particularly prone to fatigue and stress, which may be related both to school working conditions and agerelated causes (Klassen & Chiu, 2010). In the Swedish context, the declining effectiveness of veteran teachers may be explained by an increasing workload due to multiple school reforms in the recent decades. The higher workload has probably left fewer opportunities for professional development, and exacerbated teacher stress.
Our findings regarding teacher self-efficacy are in agreement with recent meta-analyses pointing to the lack of empirical support for the link between teacher self-efficacy and student achievement, particularly for secondary school students' mathematics (Klassen et al., 2011;Wyatt, 2014;Zee & Koomen, 2016). The reason for the absence of an association may be the distance between teacher self-efficacy and student achievement. The relations between the two variables follow a complex link from teacher self-efficacy to teacher behaviours, which in turn effect student beliefs and motivation, and which in their turn are connected to student achievement through student behaviours (Fives & Buehl, 2012;Tschannen-Moran et al., 1998). This intricate relational path from teacher self-efficacy to student achievement could explain the absence of a direct link between the two. The finding on a positive relationship between teacher self-efficacy beliefs and teaching experience is in line with performance accomplishments being the main source of teacher self-efficacy beliefs (Bandura, 1997).

Teacher characteristics and student-assessed instructional quality
The absence of relationships between the number of semesters of studying mathematics, teaching experience and student-assessed instructional quality was less expected, as previous research has found teachers' subject studies to be a major source of teacher CK and PCK, positively affecting teacher instructional quality and student achievement . Similarly, according to the literature, teaching experience would bridge CK and PCK and allow for further refinement of teaching skills (Friedrichsen et al., 2009). However, when instructional quality is assessed by students, it may be confounded by other factors, such as student attitudes to the subject and/or to the teacher (Wagner et al., 2013).
On the other hand, students taught by teachers with higher self-efficacy beliefs gave higher ratings of the quality of classroom instruction. This result corresponds with what has been postulated within social cognitive theory, where self-efficacy is seen as a decisive factor for effective teaching practices. It is partly in line with previous empirical research (Holzberger et al., 2013), which has demonstrated how the relation between self-efficacy and instructional quality is reciprocal. Indeed, previous experiences of successful teaching practice may also strengthen teacher beliefs about their effectivenessan idea supported by the model of reciprocal causation (Bandura, 1986).

Conclusion
The present study aimed to investigate the relations between aspects of teacher quality, student perceptions of instructional quality, and student mathematics achievement.
Results demonstrate a positive relationship between the indicator of teacher content knowledge and student mathematics achievement, after controlling for classroom SES composition. Further, our findings indicate that up until around 19 years in service, teaching experience is positively related to student achievement. Teacher self-efficacy beliefs were positively associated with instructional quality as perceived by students. However, teacher self-efficacy beliefs had no relation to the classroom mean mathematics achievement.

Limitations and further research
The current study has several limitations that should be considered. First, the crosssectional design of the TIMSS study only enables a snapshot of relationships at a certain period in time, and does not allow any conclusions about causal effects to be made. Moreover, eighth-grade students have experienced varying teacher quality during schooling, therefore teacher effects may be lagging, as well as cutting through subjects. Additionally, the relationship between certain teacher characteristics and student outcomes may be reciprocal (Holzberger et al., 2013).
Further, teacher self-efficacy beliefs, as any self-reported measure, is challenging to assess accurately. Thus findings need to be interpreted with caution. In addition, classroom management has been suggested to be an important dimension of selfefficacy. Yet, it was not represented in the TIMSS teacher self-efficacy scale, possibly leading to construct under-representation (Messick, 1995;Shadish, Cook, & Campbell, 2002). For this reason, the use of domain-and grade-specific measures of teacher selfefficacy beliefs, which cover the key dimensions of instructional, motivational and classroom management is encouraged in future research.
Similarly, the indicator of teacher content knowledge (number of semesters studying mathematics) is also likely to suffer from construct under-representation. Teachers vary in knowledge levels when they enter teacher education programs, which in turn also vary in quality and the amount of field experience offered. The translation of CK into PCK, and thus enabling effective teaching, therefore requires careful feedback from educators and mentors.
Finally, school-level factors, such as teacher working conditions, may serve as either affordances or constraints to successful teaching, and thus require further investigation.

Funding
This work was supported by The Swedish Research Council under the grant number -2207Vetenskapsrådet [2013Vetenskapsrådet [ -2207 Notes on contributor Anna Toropova is a PhD candidate at the Department of Education and Special Education at the University of Gothenburg, Sweden. Her research interests lie in the area of teacher and instructional quality as well as teacher well-being. Stefan Johansson is a senior lecturer at the Department of Education and Special Education at the University of Gothenburg, Sweden. In his previous research, he has used international largescale data to investigate different aspects of teacher characteristics on student achievement, for example, the role of teacher competence for pupils' reading literacy. Eva Myrberg is an Associate Professor at the Department of Education and Special Education at the University of Gothenburg, Sweden. She has studied causes to differences in educational outcomes and has a special interest in effects of different aspects of teacher competence on student reading achievement.