Contributed by:

Diverse stakeholders have an interest in understanding how teacher characteristics—their preparation and experience, knowledge, mindsets, and habits—relate to students’ outcomes in mathematics. Past research has extensively explored this issue but often examined each characteristic in isolation. Drawing on data from roughly 300 fourth- and fifth-grade teachers, we attend to multiple teacher characteristics and find that experience, knowledge, effort invested in noninstructional activities, and participation in mathematics content/methods courses predict student outcomes. We also find imbalances in key teacher characteristics across student populations. We discuss the implications of these findings for hiring and training mathematics teachers.

Keywords achievement, mathematics education, teacher quality, teacher qualifications.

Keywords achievement, mathematics education, teacher quality, teacher qualifications.

1.
EPXXXX10.1177/0895904818755468Educational PolicyHill et al.

Article

Educational Policy

2019, Vol. 33(7) 1103–1134

Teacher Characteristics © The Author(s) 2018

Article reuse guidelines:

and Student Learning sagepub.com/journals-permissions

DOI: 10.1177/0895904818755468

https://doi.org/10.1177/0895904818755468

in Mathematics: journals.sagepub.com/home/epx

A Comprehensive

Assessment

Heather C. Hill1, Charalambos Y. Charalambous2,

and Mark J. Chin1

Abstract

Diverse stakeholders have an interest in understanding how teacher

characteristics—their preparation and experience, knowledge, and mind-

sets and habits—relate to students’ outcomes in mathematics. Past research

has extensively explored this issue but often examined each characteristic in

isolation. Drawing on data from roughly 300 fourth- and fifth-grade teachers,

we attend to multiple teacher characteristics and find that experience,

knowledge, effort invested in noninstructional activities, and participation in

mathematics content/methods courses predict student outcomes. We also

find imbalances in key teacher characteristics across student populations. We

discuss the implications of these findings for hiring and training mathematics

teachers.

Keywords

achievement, mathematics education, teacher quality, teacher qualifications

1Harvard University, Cambridge, MA, USA

2University of Cyprus, Nicosia, Cyprus

Corresponding Author:

Mark J. Chin, Center for Education Policy Research, 50 Church Street, 4th Floor, Cambridge,

MA 02138, USA.

Email: [email protected]

Article

Educational Policy

2019, Vol. 33(7) 1103–1134

Teacher Characteristics © The Author(s) 2018

Article reuse guidelines:

and Student Learning sagepub.com/journals-permissions

DOI: 10.1177/0895904818755468

https://doi.org/10.1177/0895904818755468

in Mathematics: journals.sagepub.com/home/epx

A Comprehensive

Assessment

Heather C. Hill1, Charalambos Y. Charalambous2,

and Mark J. Chin1

Abstract

Diverse stakeholders have an interest in understanding how teacher

characteristics—their preparation and experience, knowledge, and mind-

sets and habits—relate to students’ outcomes in mathematics. Past research

has extensively explored this issue but often examined each characteristic in

isolation. Drawing on data from roughly 300 fourth- and fifth-grade teachers,

we attend to multiple teacher characteristics and find that experience,

knowledge, effort invested in noninstructional activities, and participation in

mathematics content/methods courses predict student outcomes. We also

find imbalances in key teacher characteristics across student populations. We

discuss the implications of these findings for hiring and training mathematics

teachers.

Keywords

achievement, mathematics education, teacher quality, teacher qualifications

1Harvard University, Cambridge, MA, USA

2University of Cyprus, Nicosia, Cyprus

Corresponding Author:

Mark J. Chin, Center for Education Policy Research, 50 Church Street, 4th Floor, Cambridge,

MA 02138, USA.

Email: [email protected]

2.
1104 Educational Policy 33(7)

Identifying effective teachers has become a pressing need for many state and

district policy makers. Federal legislation, such as No Child Left Behind and

Race to the Top, required local educational agencies (LEAs) to both measure

teacher quality and take action to dismiss or remediate those that fall below a

bar for sufficient quality. LEAs themselves may wish to better identify effec-

tive and ineffective teachers, as past research has shown teacher quality to be

one of the strongest institutional-level predictors of student outcomes (e.g.,

Chetty, Friedman, & Rockoff, 2014). Yet for LEAs interested in improving

the quality of teacher candidates at entry into their system, and in improving

the quality of the many teachers without student test scores and/or classroom

observations, determining teacher quality can be challenging. The same is

true for LEAs interested in strategically placing new teachers in certain

schools to more equitably distribute teacher quality.

Such LEAs might turn to the literature on the relationship between teacher

characteristics and student outcomes. The factors tested by this literature fall

into three broad categories, including the preparation and experiences hypoth-

esized to be important to teachers (e.g., postsecondary mathematics course-

work, degrees, teaching experience, certification type; Clotfelter, Ladd, &

Vigdor, 2007; Harris & Sass, 2011; Monk, 1994); teacher knowledge (e.g.,

mathematical knowledge for teaching, knowledge of students’ mathematical

abilities and misconceptions; Bell, Wilson, Higgins, & McCoach, 2010;

Carpenter, Fennema, Peterson, & Carey, 1988; Hill, Rowan, & Ball, 2005);

and teacher mind-sets and habits (e.g., efficacy, locus of control, effort

invested in teaching; Bandura, 1997; Tschannen-Moran & Hoy, 2001). Across

these categories, researchers have measured and tested numerous variables

against student learning on standardized assessments.

While many of the studies noted above show small positive associations

between measured characteristics and student outcomes, results for specific

variables are decidedly mixed across the field as a whole, making the job of

practitioners who want to take lessons from this literature difficult.

Furthermore, most studies in this genre tend to specialize in only one cate-

gory, restricting the number and type of variables tested and thus potentially

obscuring important relationships. Economists, for instance, typically exam-

ine the association between student outcomes and teacher background char-

acteristics, such as experience and degrees, by relying on district administrative

data; as this implies, such papers tend not to include predictors that require

data collection efforts. Studies of teachers’ knowledge, on the contrary, must

collect large amounts of original data to measure that knowledge, and, per-

haps as a result, rarely capture more than a handful of other key variables

(e.g., Baumert et al., 2010; Hill et al., 2005). The same holds for studies of

teacher mind-sets and habits, as well; in fact, seldom does a single study test

Identifying effective teachers has become a pressing need for many state and

district policy makers. Federal legislation, such as No Child Left Behind and

Race to the Top, required local educational agencies (LEAs) to both measure

teacher quality and take action to dismiss or remediate those that fall below a

bar for sufficient quality. LEAs themselves may wish to better identify effec-

tive and ineffective teachers, as past research has shown teacher quality to be

one of the strongest institutional-level predictors of student outcomes (e.g.,

Chetty, Friedman, & Rockoff, 2014). Yet for LEAs interested in improving

the quality of teacher candidates at entry into their system, and in improving

the quality of the many teachers without student test scores and/or classroom

observations, determining teacher quality can be challenging. The same is

true for LEAs interested in strategically placing new teachers in certain

schools to more equitably distribute teacher quality.

Such LEAs might turn to the literature on the relationship between teacher

characteristics and student outcomes. The factors tested by this literature fall

into three broad categories, including the preparation and experiences hypoth-

esized to be important to teachers (e.g., postsecondary mathematics course-

work, degrees, teaching experience, certification type; Clotfelter, Ladd, &

Vigdor, 2007; Harris & Sass, 2011; Monk, 1994); teacher knowledge (e.g.,

mathematical knowledge for teaching, knowledge of students’ mathematical

abilities and misconceptions; Bell, Wilson, Higgins, & McCoach, 2010;

Carpenter, Fennema, Peterson, & Carey, 1988; Hill, Rowan, & Ball, 2005);

and teacher mind-sets and habits (e.g., efficacy, locus of control, effort

invested in teaching; Bandura, 1997; Tschannen-Moran & Hoy, 2001). Across

these categories, researchers have measured and tested numerous variables

against student learning on standardized assessments.

While many of the studies noted above show small positive associations

between measured characteristics and student outcomes, results for specific

variables are decidedly mixed across the field as a whole, making the job of

practitioners who want to take lessons from this literature difficult.

Furthermore, most studies in this genre tend to specialize in only one cate-

gory, restricting the number and type of variables tested and thus potentially

obscuring important relationships. Economists, for instance, typically exam-

ine the association between student outcomes and teacher background char-

acteristics, such as experience and degrees, by relying on district administrative

data; as this implies, such papers tend not to include predictors that require

data collection efforts. Studies of teachers’ knowledge, on the contrary, must

collect large amounts of original data to measure that knowledge, and, per-

haps as a result, rarely capture more than a handful of other key variables

(e.g., Baumert et al., 2010; Hill et al., 2005). The same holds for studies of

teacher mind-sets and habits, as well; in fact, seldom does a single study test

3.
Hill et al. 1105

more than one class of characteristics (for exceptions, see Boonen, Van

Damme, & Onghena, 2014; Campbell et al., 2014; Grubb, 2008; Palardy &

Rumberger, 2008). Yet it seems likely that many of these factors correlate;

teachers with stronger postsecondary mathematical preparation, for instance,

may also have stronger mathematical knowledge and may feel more capable

of teaching the subject.

To untangle these relationships, we argue for a more comprehensive com-

parison of these characteristics, with the aim of understanding how they

relate to one another and, either individually or jointly, contribute to student

outcomes. Based on evidence that teacher characteristics often vary accord-

ing to the student populations they serve (e.g., Goldhaber, Lavery, &

Theobald, 2015; Hill et al., 2005; Jackson, 2009; Lankford, Loeb, & Wyckoff,

2002), we also argue for an investigation of whether key teacher characteris-

tics are distributed equitably across students and schools.

To accomplish this aim, we draw on data from roughly 300 fourth- and

fifth-grade teachers of mathematics. When examining selected teacher char-

acteristics from the three categories mentioned above, we found low to mod-

erate correlations with student outcomes, consonant with the prior literature.

Our exploration also pointed to imbalances in key teacher characteristics

across student populations. In the remainder of the article, we first review

evidence of each of the three categories of variables considered herein. After

presenting our research questions, we outline the methods we pursued in

addressing them. In the last two sections, we present the study findings and

discuss their implications for teacher hiring, placement, and training.

Teacher characteristics may contribute to student outcomes in important

ways. Background variables, such as college- or graduate-level coursework,

may demarcate experiences that provide teachers with access to curriculum

materials, classroom tasks and activities, content standards, and assessment

techniques that can shape practice and thus student outcomes. Experiences on

the job may similarly build skill in using these elements in practice. Teacher

knowledge may also inform instruction and student outcomes (Shulman,

1986); this knowledge may be acquired in coursework, in on-the-job learn-

ing, or may simply stem from differences among individuals as they enter the

profession. Teacher mind-sets and work habits may contribute to student out-

comes independently of the prior two categories; for instance, teachers may

view mathematics as a fixed set of procedures to be learned, or as a set of

practices in which to engage students, thus directly shaping their instruction.

In addition, teachers may hold different views of responsibility for learning

more than one class of characteristics (for exceptions, see Boonen, Van

Damme, & Onghena, 2014; Campbell et al., 2014; Grubb, 2008; Palardy &

Rumberger, 2008). Yet it seems likely that many of these factors correlate;

teachers with stronger postsecondary mathematical preparation, for instance,

may also have stronger mathematical knowledge and may feel more capable

of teaching the subject.

To untangle these relationships, we argue for a more comprehensive com-

parison of these characteristics, with the aim of understanding how they

relate to one another and, either individually or jointly, contribute to student

outcomes. Based on evidence that teacher characteristics often vary accord-

ing to the student populations they serve (e.g., Goldhaber, Lavery, &

Theobald, 2015; Hill et al., 2005; Jackson, 2009; Lankford, Loeb, & Wyckoff,

2002), we also argue for an investigation of whether key teacher characteris-

tics are distributed equitably across students and schools.

To accomplish this aim, we draw on data from roughly 300 fourth- and

fifth-grade teachers of mathematics. When examining selected teacher char-

acteristics from the three categories mentioned above, we found low to mod-

erate correlations with student outcomes, consonant with the prior literature.

Our exploration also pointed to imbalances in key teacher characteristics

across student populations. In the remainder of the article, we first review

evidence of each of the three categories of variables considered herein. After

presenting our research questions, we outline the methods we pursued in

addressing them. In the last two sections, we present the study findings and

discuss their implications for teacher hiring, placement, and training.

Teacher characteristics may contribute to student outcomes in important

ways. Background variables, such as college- or graduate-level coursework,

may demarcate experiences that provide teachers with access to curriculum

materials, classroom tasks and activities, content standards, and assessment

techniques that can shape practice and thus student outcomes. Experiences on

the job may similarly build skill in using these elements in practice. Teacher

knowledge may also inform instruction and student outcomes (Shulman,

1986); this knowledge may be acquired in coursework, in on-the-job learn-

ing, or may simply stem from differences among individuals as they enter the

profession. Teacher mind-sets and work habits may contribute to student out-

comes independently of the prior two categories; for instance, teachers may

view mathematics as a fixed set of procedures to be learned, or as a set of

practices in which to engage students, thus directly shaping their instruction.

In addition, teachers may hold different views of responsibility for learning

4.
1106 Educational Policy 33(7)

and expend different levels of effort toward securing positive student

Below, we review evidence connecting student mathematics learning to

teacher characteristics. We include a review of the limited studies that have

addressed multiple categories and conclude with evidence regarding the dis-

tribution of such teacher characteristics over the population of students served

by schools.

Teacher Preparation and Experience

Education production function studies have focused extensively on teachers’

preparation and experience—including the number of years taught, prepara-

tion route, degrees obtained, certification type, and postsecondary course-

work—as predictors of student outcomes. Among these variables, only

teacher experience has shown a consistent positive relationship with student

outcomes, with the gains to experience most pronounced in the early years of

teaching (Boonen et al., 2014; Chetty et al., 2014; Clotfelter et al., 2007;

Grubb, 2008; Harris & Sass, 2011; Kane, Rockoff, & Staiger, 2008; Papay &

Kraft, 2015; Rice, 2003). Teacher attainment of a bachelor’s or master’s

degree in education has mostly failed to show a relationship to student out-

comes (e.g., Clotfelter et al., 2007; Harris & Sass, 2011; Wayne & Youngs,

2003; for an exception for master’s degrees, see Guarino, Dieterle,

Bargagliotti, & Mason, 2013). Findings for other variables are mixed, includ-

ing for earned degrees (e.g., Aaronson, Barrow, & Sander, 2007; Harris &

Sass, 2011; Rowan, Correnti, & Miller, 2002) and certification (for a review,

see Cochran-Smith et al., 2012) and postsecondary mathematics content and

mathematics methods coursework (e.g., Begle, 1979; Harris & Sass, 2011;

Hill et al., 2005; Monk, 1994; Rice, 2003; Wayne & Youngs, 2003). For these

latter variables, Harris and Sass (2011) found that neither mathematics con-

tent courses nor mathematics methods courses related to student outcomes

for elementary teachers; similarly, in Hill and colleagues (2005), such courses

did not predict student outcomes for elementary teachers, once controlling

for teacher knowledge. In contrast, a research synthesis by Wayne and Youngs

(2003) found that secondary students learn more from teachers with postsec-

ondary and postgraduate coursework related to mathematics, while in Begle’s

(1979) meta-analysis, taking mathematics methods courses produced the

highest percent of positive effects of all factors examined in his work.

Notably, only a few of these studies (Chetty et al., 2014; Clotfelter et al.,

2007; Harris & Sass, 2011; Papay & Kraft, 2015) examined within-teacher

variation over time and employed designs that supported making causal

and expend different levels of effort toward securing positive student

Below, we review evidence connecting student mathematics learning to

teacher characteristics. We include a review of the limited studies that have

addressed multiple categories and conclude with evidence regarding the dis-

tribution of such teacher characteristics over the population of students served

by schools.

Teacher Preparation and Experience

Education production function studies have focused extensively on teachers’

preparation and experience—including the number of years taught, prepara-

tion route, degrees obtained, certification type, and postsecondary course-

work—as predictors of student outcomes. Among these variables, only

teacher experience has shown a consistent positive relationship with student

outcomes, with the gains to experience most pronounced in the early years of

teaching (Boonen et al., 2014; Chetty et al., 2014; Clotfelter et al., 2007;

Grubb, 2008; Harris & Sass, 2011; Kane, Rockoff, & Staiger, 2008; Papay &

Kraft, 2015; Rice, 2003). Teacher attainment of a bachelor’s or master’s

degree in education has mostly failed to show a relationship to student out-

comes (e.g., Clotfelter et al., 2007; Harris & Sass, 2011; Wayne & Youngs,

2003; for an exception for master’s degrees, see Guarino, Dieterle,

Bargagliotti, & Mason, 2013). Findings for other variables are mixed, includ-

ing for earned degrees (e.g., Aaronson, Barrow, & Sander, 2007; Harris &

Sass, 2011; Rowan, Correnti, & Miller, 2002) and certification (for a review,

see Cochran-Smith et al., 2012) and postsecondary mathematics content and

mathematics methods coursework (e.g., Begle, 1979; Harris & Sass, 2011;

Hill et al., 2005; Monk, 1994; Rice, 2003; Wayne & Youngs, 2003). For these

latter variables, Harris and Sass (2011) found that neither mathematics con-

tent courses nor mathematics methods courses related to student outcomes

for elementary teachers; similarly, in Hill and colleagues (2005), such courses

did not predict student outcomes for elementary teachers, once controlling

for teacher knowledge. In contrast, a research synthesis by Wayne and Youngs

(2003) found that secondary students learn more from teachers with postsec-

ondary and postgraduate coursework related to mathematics, while in Begle’s

(1979) meta-analysis, taking mathematics methods courses produced the

highest percent of positive effects of all factors examined in his work.

Notably, only a few of these studies (Chetty et al., 2014; Clotfelter et al.,

2007; Harris & Sass, 2011; Papay & Kraft, 2015) examined within-teacher

variation over time and employed designs that supported making causal

5.
Hill et al. 1107

Teacher Knowledge

A number of studies have empirically linked aspects of teacher mathematical

knowledge to student outcomes (for a discussion of teacher knowledge types,

see Shulman, 1986). Some studies have focused on mathematics content

knowledge in its relatively pure form, finding an association between teach-

ers’ competence in basic mathematics skills and student outcomes (e.g.,

Metzler & Woessmann, 2012; Mullens, Murnane, & Willett, 1996). Other

studies, drawing on Shulman’s (1986) framework, have shown that teachers’

pedagogical content knowledge in mathematics better predicts student learn-

ing than pure, or basic, content knowledge (Baumert et al., 2010; Campbell

et al., 2014). Still other studies have focused on measuring Ball, Thames, and

Phelps’s (2008) notion of specialized content knowledge, finding that stu-

dents perform better when their teacher has more capacity to provide mathe-

matical explanations, evaluate alternative solution methods, and visually

model the content (Hill, Kapitula, & Umland, 2011; Hill et al., 2005; Rockoff,

Jacob, Kane, & Staiger, 2011). A third category of studies has measured

teachers’ knowledge of students, including the accuracy with which teachers

can predict their students’ performance (Carpenter, Fennema, Peterson,

Chiang, & Loef, 1989; Helmke & Schrader, 1987) and the extent to which

teachers can recognize, anticipate, or interpret common student misconcep-

tions (Baumert et al., 2010; Bell et al., 2010; Carpenter et al., 1988). These

knowledge-of-students measures typically either form a composite measure

of teacher knowledge, meaning their unique effect on student learning cannot

be disentangled, or they only inconsistently predict student outcomes. From

all these studies, only one (Metzler & Woessmann, 2012) used a rigorous

design, studying within-teacher variation over two subject matters and thus

making causal claims about the effect of teacher knowledge on student

Teacher Mind-Sets and Habits

Scholars have also extensively explored teacher mind-sets and habits as pos-

sible contributors to student outcomes, with strong interest in teachers’ atti-

tudes about what constitutes disciplinary knowledge, beliefs about how

instruction should occur, and levels of enthusiasm and confidence about sub-

ject matter (Begle, 1979; Ernest, 1989; Fang, 1996). Given the growing inter-

est around intelligence and mind-sets not as fixed entities but as traits that can

be developed through effort and persistence (Dweck, 2006; Molden &

Dweck, 2006), scholars have again turned to investigating the role of mind-

sets and habits for learning, not only for students, but for teachers alike. In the

Teacher Knowledge

A number of studies have empirically linked aspects of teacher mathematical

knowledge to student outcomes (for a discussion of teacher knowledge types,

see Shulman, 1986). Some studies have focused on mathematics content

knowledge in its relatively pure form, finding an association between teach-

ers’ competence in basic mathematics skills and student outcomes (e.g.,

Metzler & Woessmann, 2012; Mullens, Murnane, & Willett, 1996). Other

studies, drawing on Shulman’s (1986) framework, have shown that teachers’

pedagogical content knowledge in mathematics better predicts student learn-

ing than pure, or basic, content knowledge (Baumert et al., 2010; Campbell

et al., 2014). Still other studies have focused on measuring Ball, Thames, and

Phelps’s (2008) notion of specialized content knowledge, finding that stu-

dents perform better when their teacher has more capacity to provide mathe-

matical explanations, evaluate alternative solution methods, and visually

model the content (Hill, Kapitula, & Umland, 2011; Hill et al., 2005; Rockoff,

Jacob, Kane, & Staiger, 2011). A third category of studies has measured

teachers’ knowledge of students, including the accuracy with which teachers

can predict their students’ performance (Carpenter, Fennema, Peterson,

Chiang, & Loef, 1989; Helmke & Schrader, 1987) and the extent to which

teachers can recognize, anticipate, or interpret common student misconcep-

tions (Baumert et al., 2010; Bell et al., 2010; Carpenter et al., 1988). These

knowledge-of-students measures typically either form a composite measure

of teacher knowledge, meaning their unique effect on student learning cannot

be disentangled, or they only inconsistently predict student outcomes. From

all these studies, only one (Metzler & Woessmann, 2012) used a rigorous

design, studying within-teacher variation over two subject matters and thus

making causal claims about the effect of teacher knowledge on student

Teacher Mind-Sets and Habits

Scholars have also extensively explored teacher mind-sets and habits as pos-

sible contributors to student outcomes, with strong interest in teachers’ atti-

tudes about what constitutes disciplinary knowledge, beliefs about how

instruction should occur, and levels of enthusiasm and confidence about sub-

ject matter (Begle, 1979; Ernest, 1989; Fang, 1996). Given the growing inter-

est around intelligence and mind-sets not as fixed entities but as traits that can

be developed through effort and persistence (Dweck, 2006; Molden &

Dweck, 2006), scholars have again turned to investigating the role of mind-

sets and habits for learning, not only for students, but for teachers alike. In the

6.
1108 Educational Policy 33(7)

latter case, scholars have taken up whether teachers believe in their capacities

to influence the learning of all their students through their teaching. One such

construct is perceived efficacy, defined as teachers’ perceptions of their abil-

ity to organize and execute teaching that promotes learning (Bandura, 1997;

Charalambous, Philippou, & Kyriakides, 2008). Teacher efficacy beliefs

have consistently positively related to teachers’ behavior in the classroom

and the quality of their instruction (Justice, Mashburn, Hamre, & Pianta,

2008; Stipek, 2012; Tschannen-Moran & Hoy, 2001; Tschannen-Moran, Hoy,

& Hoy, 1998). They have also predicted students’ learning outcomes, both

cognitive (e.g., Palardy & Rumberger, 2008) and affective (e.g., Anderson,

Greene, & Loewen, 1988; Soodak & Podell, 1996).

Teacher locus of control taps the extent to which teachers feel they can

influence their students’ outcomes or whether, alternatively, they believe that

those outcomes mostly hinge on non-classroom and school factors (e.g., stu-

dents’ socioeconomic background and parental support). Drawing on Rotter’s

(1966) work on internal versus external control of reinforcement, researchers

have explored the relationship between teachers’ locus of control and student

learning, often documenting a positive relationship between the two con-

structs (Berman & McLaughlin, 1977; Rose & Medway, 1981).

A central tenet in the growth mind-set theory is the role that effort plays in

improvement. However, scholars have studied teacher effort less extensively

than efficacy and locus of control. In the only study in this category that used

a research design that allowed for making causal inferences, Lavy (2009)

observed that teacher effort mediates the positive impact of teacher merit pay

on students’ outcomes; survey data suggest that after-school tutoring plays a

strong role, in particular, in producing improved outcomes. However, to our

knowledge, few other studies have captured this variable.

Studies Measuring Multiple Categories

Within each of the three categories above, scholars have identified teacher

characteristics that significantly and positively predict student outcomes.

However, we could locate only four studies that examine the relationship

between student outcomes and variables in more than one category; in all

these studies, characteristics from different categories were concurrently fit

into the same model(s). Investigating teacher preparation and experience,

teacher attitudes, and self-reported practices, Palardy and Rumberger (2008)

found that some self-reported instructional practices and attitudes (i.e., effi-

cacy) predicted student outcomes, but teacher background characteristics did

not. In Boonen and colleagues’ (2014) work, teacher experience and job sat-

isfaction—a background characteristic and attitude, respectively—predicted

latter case, scholars have taken up whether teachers believe in their capacities

to influence the learning of all their students through their teaching. One such

construct is perceived efficacy, defined as teachers’ perceptions of their abil-

ity to organize and execute teaching that promotes learning (Bandura, 1997;

Charalambous, Philippou, & Kyriakides, 2008). Teacher efficacy beliefs

have consistently positively related to teachers’ behavior in the classroom

and the quality of their instruction (Justice, Mashburn, Hamre, & Pianta,

2008; Stipek, 2012; Tschannen-Moran & Hoy, 2001; Tschannen-Moran, Hoy,

& Hoy, 1998). They have also predicted students’ learning outcomes, both

cognitive (e.g., Palardy & Rumberger, 2008) and affective (e.g., Anderson,

Greene, & Loewen, 1988; Soodak & Podell, 1996).

Teacher locus of control taps the extent to which teachers feel they can

influence their students’ outcomes or whether, alternatively, they believe that

those outcomes mostly hinge on non-classroom and school factors (e.g., stu-

dents’ socioeconomic background and parental support). Drawing on Rotter’s

(1966) work on internal versus external control of reinforcement, researchers

have explored the relationship between teachers’ locus of control and student

learning, often documenting a positive relationship between the two con-

structs (Berman & McLaughlin, 1977; Rose & Medway, 1981).

A central tenet in the growth mind-set theory is the role that effort plays in

improvement. However, scholars have studied teacher effort less extensively

than efficacy and locus of control. In the only study in this category that used

a research design that allowed for making causal inferences, Lavy (2009)

observed that teacher effort mediates the positive impact of teacher merit pay

on students’ outcomes; survey data suggest that after-school tutoring plays a

strong role, in particular, in producing improved outcomes. However, to our

knowledge, few other studies have captured this variable.

Studies Measuring Multiple Categories

Within each of the three categories above, scholars have identified teacher

characteristics that significantly and positively predict student outcomes.

However, we could locate only four studies that examine the relationship

between student outcomes and variables in more than one category; in all

these studies, characteristics from different categories were concurrently fit

into the same model(s). Investigating teacher preparation and experience,

teacher attitudes, and self-reported practices, Palardy and Rumberger (2008)

found that some self-reported instructional practices and attitudes (i.e., effi-

cacy) predicted student outcomes, but teacher background characteristics did

not. In Boonen and colleagues’ (2014) work, teacher experience and job sat-

isfaction—a background characteristic and attitude, respectively—predicted

7.
Hill et al. 1109

Flemish students’ mathematics outcomes. Grubb (2008) reported positive

relationships between a variety of teacher background and preparation char-

acteristics (e.g., experience, teaching in-field, education track), teacher effi-

cacy beliefs, and student outcomes in mathematics in the NELS:88 data.

Finally, Campbell et al. (2014) found that teacher knowledge positively asso-

ciated with student outcomes, special education certification negatively asso-

ciated with those outcomes, and teacher attitudes and beliefs largely had no

effects outside interactions with knowledge itself.

Studies Examining the Distribution of Teacher Characteristics

Across Students

Schools serving higher proportions of non-White and impoverished students

have traditionally employed less qualified teachers, where qualified has been

defined as individuals who are fully certified, hold advanced degrees, have

prior teaching experience, and score better on certification exams or other

standardized teaching-related exams (Hill and Lubienski, 2007; Jackson,

2009; Lankford et al., 2002). Such studies often used U.S. state or national-

level data to examine teacher sorting; scholars have less often studied sorting

within urban districts or metropolitan areas. Choi (2010) demonstrated that

disadvantaged minorities and free/reduced-price lunch (FRPL) recipients in

the Los Angeles Unified School District received instruction from, on aver-

age, less qualified teachers. Schultz (2014) replicated this result for the St.

Louis metropolitan area, while a recent report from the Organization of

Economic Co-Operation and Development (Schleicher & Organisation for

Economic Co-Operation and Development, 2014) corroborated this finding

internationally, reporting higher concentrations of unqualified teachers in

schools serving disadvantaged students in several countries, including

Belgium, Chile, the Czech Republic, Iceland, Luxembourg, the Netherlands,

and Slovenia.

Synthesizing results from three studies that collectively examined the dis-

tribution of effective teachers across schools and districts across 17 U.S.

states, Max and Glazerman (2014) have also reported that, on average, disad-

vantaged students in Grades 3 to 8 receive less effective teaching in mathe-

matics than their counterparts. This difference amounted to 2 weeks of

learning (which was calculated to be equivalent to 2%-3% of the achieve-

ment gap between disadvantaged and nondisadvantaged students) and varied

across districts from less than a week in some districts to almost 8 weeks in

others. Middle-grade students in the lowest poverty schools were addition-

ally twice as likely to get teachers with value-added scores in the top 20% of

their district compared with their counterparts in the highest poverty schools.

Flemish students’ mathematics outcomes. Grubb (2008) reported positive

relationships between a variety of teacher background and preparation char-

acteristics (e.g., experience, teaching in-field, education track), teacher effi-

cacy beliefs, and student outcomes in mathematics in the NELS:88 data.

Finally, Campbell et al. (2014) found that teacher knowledge positively asso-

ciated with student outcomes, special education certification negatively asso-

ciated with those outcomes, and teacher attitudes and beliefs largely had no

effects outside interactions with knowledge itself.

Studies Examining the Distribution of Teacher Characteristics

Across Students

Schools serving higher proportions of non-White and impoverished students

have traditionally employed less qualified teachers, where qualified has been

defined as individuals who are fully certified, hold advanced degrees, have

prior teaching experience, and score better on certification exams or other

standardized teaching-related exams (Hill and Lubienski, 2007; Jackson,

2009; Lankford et al., 2002). Such studies often used U.S. state or national-

level data to examine teacher sorting; scholars have less often studied sorting

within urban districts or metropolitan areas. Choi (2010) demonstrated that

disadvantaged minorities and free/reduced-price lunch (FRPL) recipients in

the Los Angeles Unified School District received instruction from, on aver-

age, less qualified teachers. Schultz (2014) replicated this result for the St.

Louis metropolitan area, while a recent report from the Organization of

Economic Co-Operation and Development (Schleicher & Organisation for

Economic Co-Operation and Development, 2014) corroborated this finding

internationally, reporting higher concentrations of unqualified teachers in

schools serving disadvantaged students in several countries, including

Belgium, Chile, the Czech Republic, Iceland, Luxembourg, the Netherlands,

and Slovenia.

Synthesizing results from three studies that collectively examined the dis-

tribution of effective teachers across schools and districts across 17 U.S.

states, Max and Glazerman (2014) have also reported that, on average, disad-

vantaged students in Grades 3 to 8 receive less effective teaching in mathe-

matics than their counterparts. This difference amounted to 2 weeks of

learning (which was calculated to be equivalent to 2%-3% of the achieve-

ment gap between disadvantaged and nondisadvantaged students) and varied

across districts from less than a week in some districts to almost 8 weeks in

others. Middle-grade students in the lowest poverty schools were addition-

ally twice as likely to get teachers with value-added scores in the top 20% of

their district compared with their counterparts in the highest poverty schools.

8.
1110 Educational Policy 33(7)

In a more recent study, Goldhaber et al. (2015) examined this issue more

comprehensively by using a set of indicators of teacher quality—experience,

licensure exam scores, and value-added estimates of effectiveness—across a

set of indicators of student disadvantage, including FRPL status, underrepre-

sented minority, and low prior academic performance. Although focusing on

just one U.S. state, their study showed unequal distribution of almost every

single indicator of teaching quality across every indicator of student disad-

vantage, a pattern that held for every school level examined (elementary,

middle school, and high school). Much of this inequitable distribution of

teaching quality appears to be due to teacher and student sorting across dis-

tricts and schools rather than to inequitable distribution across classrooms

within schools. This key role of the teacher and student sorting across dis-

tricts appears in a recent brief (Goldhaber, Quince, & Theobald, 2016) that

searched for explanations for the different estimates of the teacher quality

gaps—namely, the unequal distribution of effective teachers across stu-

dents—reported in studies conducted from 2013 to 2016. Regardless of the

specifications of the value-added model employed in these studies, the sub-

ject, and the grade-level examined, districts that served more disadvantaged

students tended to have lower average teacher quality. Collectively, these

studies highlight not only the unequal distribution of teacher quality among

disadvantaged and nondisadvantaged students but also pinpoint the sorting of

students across districts as a key contributor to these inequalities.

Research Questions

Our research extends the knowledge base described above by simultaneously

using multiple indicators of teacher characteristics and experiences to predict

student mathematics outcomes. We do so because many extant studies

explore only a small number of variables and measures; this leads to the pos-

sibility of omitted variable bias and the misidentification of important char-

acteristics. Furthermore, even the studies reviewed above that contain more

than one type of teacher indicator typically do not report the relationships

between teacher characteristics.

We focus exclusively on teacher characteristics even though we would

typically expect that more proximal measures to student learning (i.e.,

instructional quality) would best predict student learning in mathematics.

Focusing on characteristics, however, asks a different question: What kinds

of knowledge and experiences might be associated with successful teaching?

Answering this question provides information to agencies seeking to hire

teachers in ways that maximize student outcomes, and to place teachers

within districts in equitable ways. Specifically, we ask the following research

In a more recent study, Goldhaber et al. (2015) examined this issue more

comprehensively by using a set of indicators of teacher quality—experience,

licensure exam scores, and value-added estimates of effectiveness—across a

set of indicators of student disadvantage, including FRPL status, underrepre-

sented minority, and low prior academic performance. Although focusing on

just one U.S. state, their study showed unequal distribution of almost every

single indicator of teaching quality across every indicator of student disad-

vantage, a pattern that held for every school level examined (elementary,

middle school, and high school). Much of this inequitable distribution of

teaching quality appears to be due to teacher and student sorting across dis-

tricts and schools rather than to inequitable distribution across classrooms

within schools. This key role of the teacher and student sorting across dis-

tricts appears in a recent brief (Goldhaber, Quince, & Theobald, 2016) that

searched for explanations for the different estimates of the teacher quality

gaps—namely, the unequal distribution of effective teachers across stu-

dents—reported in studies conducted from 2013 to 2016. Regardless of the

specifications of the value-added model employed in these studies, the sub-

ject, and the grade-level examined, districts that served more disadvantaged

students tended to have lower average teacher quality. Collectively, these

studies highlight not only the unequal distribution of teacher quality among

disadvantaged and nondisadvantaged students but also pinpoint the sorting of

students across districts as a key contributor to these inequalities.

Research Questions

Our research extends the knowledge base described above by simultaneously

using multiple indicators of teacher characteristics and experiences to predict

student mathematics outcomes. We do so because many extant studies

explore only a small number of variables and measures; this leads to the pos-

sibility of omitted variable bias and the misidentification of important char-

acteristics. Furthermore, even the studies reviewed above that contain more

than one type of teacher indicator typically do not report the relationships

between teacher characteristics.

We focus exclusively on teacher characteristics even though we would

typically expect that more proximal measures to student learning (i.e.,

instructional quality) would best predict student learning in mathematics.

Focusing on characteristics, however, asks a different question: What kinds

of knowledge and experiences might be associated with successful teaching?

Answering this question provides information to agencies seeking to hire

teachers in ways that maximize student outcomes, and to place teachers

within districts in equitable ways. Specifically, we ask the following research

9.
Hill et al. 1111

Research Question 1: How do measures describing teacher preparation

and experience, teacher knowledge, and teacher mind-sets and habits cor-

relate with one another and relate to student outcomes in mathematics

when concurrently examined?

Research Question 2: How are key teacher characteristics distributed

across student populations within districts?

Because our dataset includes two different kinds of mathematics tests—stan-

dardized state tests as well as a content-aligned project-administered test—

we can examine the consistency of our findings, important given prior

research reporting divergent results across different assessments (Lockwood

et al., 2007; Papay, 2011).

Data and Methods

Our data come from the National Center for Teacher Effectiveness main

study, which spanned three academic years, from 2010-2011 to 2012-2013.

The study, which developed and validated several measures of mathematics

teacher effectiveness, collected data from fourth- and fifth-grade teachers and

their students in four large urban East Coast public school districts. The proj-

ect recruited 583 teachers across the four districts, of which 328 matriculated

into the study. After excluding majority special education classrooms and

those with excessive missing student data at student baseline assessment, we

arrived at an analytic sample of 306 teachers and 10,233 students over the

three study years.

Our teacher sample was generally experienced, with an average self-

reported 10.22 years (SD = 7.23 years) in teaching at entry into our study.

Most of the sample was traditionally certified (86%), and roughly half had a

bachelor’s degree in education (53%). A small proportion had a mathematics-

specific certification (15%), and a relatively large fraction reported possess-

ing a master’s degree (76%). Student demographics reflected those in most

urban settings, with 64% of students eligible for FRPL, 10% qualified for

special education (SPED), and 20% designated as English language learners

(ELLs) at the time of the study. A notable percentage of the participating

students were either Black (40%) or Hispanic (23%).

Data Sources and Reduction

Data collection relied upon several instruments, including a background and

experience questionnaire administered once to teachers in their first year of

Research Question 1: How do measures describing teacher preparation

and experience, teacher knowledge, and teacher mind-sets and habits cor-

relate with one another and relate to student outcomes in mathematics

when concurrently examined?

Research Question 2: How are key teacher characteristics distributed

across student populations within districts?

Because our dataset includes two different kinds of mathematics tests—stan-

dardized state tests as well as a content-aligned project-administered test—

we can examine the consistency of our findings, important given prior

research reporting divergent results across different assessments (Lockwood

et al., 2007; Papay, 2011).

Data and Methods

Our data come from the National Center for Teacher Effectiveness main

study, which spanned three academic years, from 2010-2011 to 2012-2013.

The study, which developed and validated several measures of mathematics

teacher effectiveness, collected data from fourth- and fifth-grade teachers and

their students in four large urban East Coast public school districts. The proj-

ect recruited 583 teachers across the four districts, of which 328 matriculated

into the study. After excluding majority special education classrooms and

those with excessive missing student data at student baseline assessment, we

arrived at an analytic sample of 306 teachers and 10,233 students over the

three study years.

Our teacher sample was generally experienced, with an average self-

reported 10.22 years (SD = 7.23 years) in teaching at entry into our study.

Most of the sample was traditionally certified (86%), and roughly half had a

bachelor’s degree in education (53%). A small proportion had a mathematics-

specific certification (15%), and a relatively large fraction reported possess-

ing a master’s degree (76%). Student demographics reflected those in most

urban settings, with 64% of students eligible for FRPL, 10% qualified for

special education (SPED), and 20% designated as English language learners

(ELLs) at the time of the study. A notable percentage of the participating

students were either Black (40%) or Hispanic (23%).

Data Sources and Reduction

Data collection relied upon several instruments, including a background and

experience questionnaire administered once to teachers in their first year of

10.
1112 Educational Policy 33(7)

study participation; a fall questionnaire, administered each school year and

comprising questions measuring teachers’ mathematical knowledge as well

as questions related to teachers’ mind-sets and habits; and a spring question-

naire, administered each school year and containing items assessing teachers’

knowledge of students. As noted above, we gauged student performance with

both a project-developed mathematics assessment (see Hickman, Fu, & Hill,

2012) and with state standardized test scores. Districts provided the latter

scores in addition to student demographic information.

Below, we describe the teacher measures used in this study. We selected

these measures, which we organize into the three categories described in our

literature review, based on prior theoretical and empirical evidence support-

ing their importance for student learning.

Teacher Preparation and Experience Measures

We used teacher responses to eight survey items to develop the following

measures related to preparation and experience:

•• A dichotomous variable indicating novice teachers (i.e., those with no

more than 2 years of experience);

•• Ordinal variables, with responses ranging from 1 (“no classes”) to 4

(“six or more classes”), indicating teachers’ reported number of under-

graduate or graduate-level classes covering college-level mathematics

topics (mathematics courses), mathematics content for teachers (math-

ematics content courses), and methods for teaching mathematics

(mathematics methods courses);

•• A dichotomous variable indicating a traditional pathway into the pro-

fession (traditionally certified) as opposed to participation in an alter-

native certification program (e.g., Teach for America) or no

participation in any formal training;

•• A dichotomous variable indicating possession of a bachelor’s degree

in education;

•• A dichotomous variable indicating possession of a certificate in the

teaching of elementary mathematics; and

•• A dichotomous variable indicating possession of any master’s degree.

Teacher Knowledge Measures

The fall and spring teacher questionnaires supplied two measures of teacher

knowledge. The first was MKT/STEL, built from items assessing teachers’

Mathematical Knowledge for Teaching (MKT; Hill et al., 2005) and items

study participation; a fall questionnaire, administered each school year and

comprising questions measuring teachers’ mathematical knowledge as well

as questions related to teachers’ mind-sets and habits; and a spring question-

naire, administered each school year and containing items assessing teachers’

knowledge of students. As noted above, we gauged student performance with

both a project-developed mathematics assessment (see Hickman, Fu, & Hill,

2012) and with state standardized test scores. Districts provided the latter

scores in addition to student demographic information.

Below, we describe the teacher measures used in this study. We selected

these measures, which we organize into the three categories described in our

literature review, based on prior theoretical and empirical evidence support-

ing their importance for student learning.

Teacher Preparation and Experience Measures

We used teacher responses to eight survey items to develop the following

measures related to preparation and experience:

•• A dichotomous variable indicating novice teachers (i.e., those with no

more than 2 years of experience);

•• Ordinal variables, with responses ranging from 1 (“no classes”) to 4

(“six or more classes”), indicating teachers’ reported number of under-

graduate or graduate-level classes covering college-level mathematics

topics (mathematics courses), mathematics content for teachers (math-

ematics content courses), and methods for teaching mathematics

(mathematics methods courses);

•• A dichotomous variable indicating a traditional pathway into the pro-

fession (traditionally certified) as opposed to participation in an alter-

native certification program (e.g., Teach for America) or no

participation in any formal training;

•• A dichotomous variable indicating possession of a bachelor’s degree

in education;

•• A dichotomous variable indicating possession of a certificate in the

teaching of elementary mathematics; and

•• A dichotomous variable indicating possession of any master’s degree.

Teacher Knowledge Measures

The fall and spring teacher questionnaires supplied two measures of teacher

knowledge. The first was MKT/STEL, built from items assessing teachers’

Mathematical Knowledge for Teaching (MKT; Hill et al., 2005) and items

11.
Hill et al. 1113

from a state Test of Education Licensure (STEL). We originally hoped these

two types of items would form two measures, one representing teachers’

basic mathematical competence (STEL) and one representing teachers’ spe-

cialized mathematical knowledge (MKT), but a factor analysis suggested

these dimensions could not be distinguished (Charalambous, Hill, McGinn &

Chin, 2017, manuscript in preparation). The second knowledge measure

tapped teachers’ accuracy in predicting student performance on items from

the project test. We presented teachers with items from the project-adminis-

tered mathematics assessment, and then asked what percent of their students

would answer the item correctly. Using these data, we calculated the absolute

difference between the teacher estimate and actual percentage correct within

the teacher’s classroom. We then estimated a multilevel model, with each dif-

ference as the dependent variable, that crossed fixed item effects with random

teacher effects while including weights for the number of students in each

classroom; we adjusted the random effects from this model—the accuracy

scores—for the classroom composition of students on evidence that teachers

of low-performing students may receive higher difference scores because

teachers are generally overoptimistic regarding student outcomes rather than

a true difference in accuracy (see Hill & Chin, under review). The MKT/

STEL measure possessed a reliability of .92; the adjusted intraclass correla-

tions of the teacher accuracy scores ranged from .71 to .79.1 For more infor-

mation on the construction and validity of these knowledge measures, please

see (Hill & Chin, under review).

Teacher Mind-Sets and Habits Measures

Responses to items on the fall questionnaire allowed us to estimate scores

reflecting teacher efficacy, locus of control, and effort invested in noninstruc-

tional activities. We adapted efficacy items from Tschannen-Moran and Hoy

(2001); these items describe teachers’ assessment of their ability to carry out

common classroom activities (e.g., crafting good questions). We selected

locus of control items from Hoy and Woolfolk (1993) and Dweck, Chiu, and

Hong (1995); these items capture teacher beliefs about, for example, whether

or not students can change their intelligence or learn new things. Project staff

created effort items, which captured the amount of time spent on noninstruc-

tional activities like grading homework or securing resources for students.

Efficacy and locus of control were subject-independent metrics, whereas the

effort measure was specifically tied to mathematics. Table 1 shows the items

related to each measure and the internal consistencies of composites. Because

the study asked teachers about efficacy and effort in multiple years, we lever-

aged this additional information using the following equation:

from a state Test of Education Licensure (STEL). We originally hoped these

two types of items would form two measures, one representing teachers’

basic mathematical competence (STEL) and one representing teachers’ spe-

cialized mathematical knowledge (MKT), but a factor analysis suggested

these dimensions could not be distinguished (Charalambous, Hill, McGinn &

Chin, 2017, manuscript in preparation). The second knowledge measure

tapped teachers’ accuracy in predicting student performance on items from

the project test. We presented teachers with items from the project-adminis-

tered mathematics assessment, and then asked what percent of their students

would answer the item correctly. Using these data, we calculated the absolute

difference between the teacher estimate and actual percentage correct within

the teacher’s classroom. We then estimated a multilevel model, with each dif-

ference as the dependent variable, that crossed fixed item effects with random

teacher effects while including weights for the number of students in each

classroom; we adjusted the random effects from this model—the accuracy

scores—for the classroom composition of students on evidence that teachers

of low-performing students may receive higher difference scores because

teachers are generally overoptimistic regarding student outcomes rather than

a true difference in accuracy (see Hill & Chin, under review). The MKT/

STEL measure possessed a reliability of .92; the adjusted intraclass correla-

tions of the teacher accuracy scores ranged from .71 to .79.1 For more infor-

mation on the construction and validity of these knowledge measures, please

see (Hill & Chin, under review).

Teacher Mind-Sets and Habits Measures

Responses to items on the fall questionnaire allowed us to estimate scores

reflecting teacher efficacy, locus of control, and effort invested in noninstruc-

tional activities. We adapted efficacy items from Tschannen-Moran and Hoy

(2001); these items describe teachers’ assessment of their ability to carry out

common classroom activities (e.g., crafting good questions). We selected

locus of control items from Hoy and Woolfolk (1993) and Dweck, Chiu, and

Hong (1995); these items capture teacher beliefs about, for example, whether

or not students can change their intelligence or learn new things. Project staff

created effort items, which captured the amount of time spent on noninstruc-

tional activities like grading homework or securing resources for students.

Efficacy and locus of control were subject-independent metrics, whereas the

effort measure was specifically tied to mathematics. Table 1 shows the items

related to each measure and the internal consistencies of composites. Because

the study asked teachers about efficacy and effort in multiple years, we lever-

aged this additional information using the following equation:

12.
1114 Educational Policy 33(7)

Table 1. Descriptions of Teacher Mind-Sets and Habits Measures.

Alpha Alpha

Items 2010-2011 2011-2012

Efficacya Belief in ability to, for example, craft .66 .86

good questions for students, provide

alternative explanations or examples

to confused students, use a variety of

assessment strategies to help students

learn, control disruptive behavior

Locus of Belief in, for example, whether or not .93

control students can change their intelligence;

whether or not students learn new things

Effort Time spent per week, for example, on .79 .73

grading math assignments, gathering

and organizing math lesson material,

reviewing the content of specific math

lessons, helping students learn any

subject after school hours

aFor the efficacy measure, items and scales changed between 2010-2011 and 2011-2012.

Scores are thus standardized within school year.

TQ yt = β0 + α y + µt + ε yt (1)

The outcome, TQ yt, represents the average of teacher t’s responses, within

year y, across the items of each respective construct. The model controls for

differences in average response level across years using year fixed effects,

α y . The random effect for teacher t, µt, comprises each teacher’s score on

effort or efficacy.2

Student Mathematics Tests

We employed two measures of student learning in mathematics. First, dis-

tricts supplied student scores on state mathematics tests for the years of the

study and for up to 2 years prior; many schools and teachers experienced

these as high-stakes assessments due to No Child Left Behind regulations.

These state tests ranged in content, from two that primarily focused on basic

skills and problem-solving to one—used in two study districts—that required

more complex thinking and communication about mathematics (Lynch, Chin

& Blazer, 2017). Second, sampled students completed a project-developed

mathematics test in the spring semester of each school year. Project staff

designed this assessment in partnership with the Educational Testing Service

Table 1. Descriptions of Teacher Mind-Sets and Habits Measures.

Alpha Alpha

Items 2010-2011 2011-2012

Efficacya Belief in ability to, for example, craft .66 .86

good questions for students, provide

alternative explanations or examples

to confused students, use a variety of

assessment strategies to help students

learn, control disruptive behavior

Locus of Belief in, for example, whether or not .93

control students can change their intelligence;

whether or not students learn new things

Effort Time spent per week, for example, on .79 .73

grading math assignments, gathering

and organizing math lesson material,

reviewing the content of specific math

lessons, helping students learn any

subject after school hours

aFor the efficacy measure, items and scales changed between 2010-2011 and 2011-2012.

Scores are thus standardized within school year.

TQ yt = β0 + α y + µt + ε yt (1)

The outcome, TQ yt, represents the average of teacher t’s responses, within

year y, across the items of each respective construct. The model controls for

differences in average response level across years using year fixed effects,

α y . The random effect for teacher t, µt, comprises each teacher’s score on

effort or efficacy.2

Student Mathematics Tests

We employed two measures of student learning in mathematics. First, dis-

tricts supplied student scores on state mathematics tests for the years of the

study and for up to 2 years prior; many schools and teachers experienced

these as high-stakes assessments due to No Child Left Behind regulations.

These state tests ranged in content, from two that primarily focused on basic

skills and problem-solving to one—used in two study districts—that required

more complex thinking and communication about mathematics (Lynch, Chin

& Blazer, 2017). Second, sampled students completed a project-developed

mathematics test in the spring semester of each school year. Project staff

designed this assessment in partnership with the Educational Testing Service

13.
Hill et al. 1115

to include cognitively challenging and mathematically complex problems.

The staff hoped that the assessment would prove more reflective of current

standards for student learning (i.e., Common Core Standards for Mathematics)

and would more strongly align to the study’s mathematics-specific knowl-

edge measures. Student-level reliabilities for this test ranged from .82 to .89.

Depending on the academic year, student-level correlations between the state

and project tests ranged from .69 to .75 and correlations of teacher value-

added scores based on these tests ranged from .29 to .53.

Scoring and Imputation of Missing Data

For ease of interpretation, we standardized students’ test scores. Specifically,

we standardized students’ project-developed mathematics test scores across

districts to have a mean of zero and an SD of 1; we similarly standardized

state mathematics test scores, but did so within district because the assess-

ments differed across each context. We de-meaned the teacher scores

described above by district to account for the sometimes sizable differences

in teacher characteristics across those districts. Doing so also mirrored the

scoring of students’ state standardized test performance, which was standard-

ized within district to account for differences in tests. Roughly 63% of the

teachers used in our analyses had complete data; 95% of teachers were miss-

ing four variables at most. For cases of missing data, we imputed scores using

the district mean and included a dichotomous indicator designating whether

a teacher was missing data from a specific source (i.e., from the background,

fall, or spring questionnaire).

Analysis Strategies

We used three primary strategies in our analyses. We began by correlating

teacher scores from all the measures listed above. Doing so describes the

relationship of the teacher characteristics to one another and also serves as a

check for multicollinearity. Next, we predicted student performance on both

the state and project mathematics tests using the following multilevel model,

which nests students within teacher–year combinations, which are subse-

quently nested within teacher:

yspcgyt = β0 + αX sy −1 + δDsy + φPpcgyt + κCcgyt + η + ωθt + µt + ν yt + ε spcgyt (2)

The outcome, yspcgyt , represents either the state or project test performance

of student s, in classroom p, in cohort (i.e., school, year, and grade) c, taking

the test for grade g, in year y, taught by teacher t. Equation 2 contains the fol-

lowing controls:

to include cognitively challenging and mathematically complex problems.

The staff hoped that the assessment would prove more reflective of current

standards for student learning (i.e., Common Core Standards for Mathematics)

and would more strongly align to the study’s mathematics-specific knowl-

edge measures. Student-level reliabilities for this test ranged from .82 to .89.

Depending on the academic year, student-level correlations between the state

and project tests ranged from .69 to .75 and correlations of teacher value-

added scores based on these tests ranged from .29 to .53.

Scoring and Imputation of Missing Data

For ease of interpretation, we standardized students’ test scores. Specifically,

we standardized students’ project-developed mathematics test scores across

districts to have a mean of zero and an SD of 1; we similarly standardized

state mathematics test scores, but did so within district because the assess-

ments differed across each context. We de-meaned the teacher scores

described above by district to account for the sometimes sizable differences

in teacher characteristics across those districts. Doing so also mirrored the

scoring of students’ state standardized test performance, which was standard-

ized within district to account for differences in tests. Roughly 63% of the

teachers used in our analyses had complete data; 95% of teachers were miss-

ing four variables at most. For cases of missing data, we imputed scores using

the district mean and included a dichotomous indicator designating whether

a teacher was missing data from a specific source (i.e., from the background,

fall, or spring questionnaire).

Analysis Strategies

We used three primary strategies in our analyses. We began by correlating

teacher scores from all the measures listed above. Doing so describes the

relationship of the teacher characteristics to one another and also serves as a

check for multicollinearity. Next, we predicted student performance on both

the state and project mathematics tests using the following multilevel model,

which nests students within teacher–year combinations, which are subse-

quently nested within teacher:

yspcgyt = β0 + αX sy −1 + δDsy + φPpcgyt + κCcgyt + η + ωθt + µt + ν yt + ε spcgyt (2)

The outcome, yspcgyt , represents either the state or project test performance

of student s, in classroom p, in cohort (i.e., school, year, and grade) c, taking

the test for grade g, in year y, taught by teacher t. Equation 2 contains the fol-

lowing controls:

14.
1116 Educational Policy 33(7)

•• X sy −1 , a vector of controls for student prior test performance;

•• Dsy , a vector of controls for student demographic information (i.e.,

race or ethnicity, gender, FRPL-eligibility; SPED status; and ELL

status);

•• Ppcgyt , classroom-level averages of X sy −1 and Dsy to capture the effects

of a student’s peers;

•• Ccgyt , cohort-level averages of X sy−1 and Dsy to capture the effect of

a student’s cohort;

•• η , district and grade-by-year fixed effects;

•• θt , a vector of teacher-level scores for teacher characteristic measures3;

•• µt, a random effect on test performance for being taught by teacher t ;

and

•• ν yt , a random effect on test performance for being taught by teacher t

in year y.

The model represented by Equation 2 contains controls used by many states

and districts when estimating teacher value-added scores. One advantage of

this model is that it uses multiple classroom-years to construct teacher value-

added scores; prior research shows that doing so results in less biased and

more reliable score estimates (Goldhaber & Hansen, 2013; Koedel & Betts,

2011). We included classroom- and cohort-average student demographics as

well as district fixed effects to account for the sorting of teachers to student

populations. As our data show evidence of such sorting (discussed below), we

considered these controls appropriate; to further control for sorting, we also

conducted specification checks including models with school fixed effects.

Results (available upon request) largely replicated those presented here.

We recovered our primary parameters of interest, coefficients representing

the relationship between specific teacher characteristics to student test per-

formance, from ω . To make interpretation of these coefficients easier, we

standardized teacher scores on the mathematics courses, knowledge, and

mind-sets and habits measures across the teacher sample prior to model esti-

mation. We estimated Equation 2 four times for each outcome, testing the

variables first by category and then in an omnibus model. As we had a rela-

tively large number of variables given our sample size, we set a slightly

higher threshold for statistical significance, referring to estimates with p val-

ues between .10 and .05 as marginally significant. For each model estimation,

we conducted a Wald test to examine the joint significance of each category’s

variables in predicting outcomes and also reported the amount of variance in

teacher effects explained by each category (an “adjusted pseudo R-squared”;

see Bacher-Hicks, Chin, Hill & Straiger, 2018).

•• X sy −1 , a vector of controls for student prior test performance;

•• Dsy , a vector of controls for student demographic information (i.e.,

race or ethnicity, gender, FRPL-eligibility; SPED status; and ELL

status);

•• Ppcgyt , classroom-level averages of X sy −1 and Dsy to capture the effects

of a student’s peers;

•• Ccgyt , cohort-level averages of X sy−1 and Dsy to capture the effect of

a student’s cohort;

•• η , district and grade-by-year fixed effects;

•• θt , a vector of teacher-level scores for teacher characteristic measures3;

•• µt, a random effect on test performance for being taught by teacher t ;

and

•• ν yt , a random effect on test performance for being taught by teacher t

in year y.

The model represented by Equation 2 contains controls used by many states

and districts when estimating teacher value-added scores. One advantage of

this model is that it uses multiple classroom-years to construct teacher value-

added scores; prior research shows that doing so results in less biased and

more reliable score estimates (Goldhaber & Hansen, 2013; Koedel & Betts,

2011). We included classroom- and cohort-average student demographics as

well as district fixed effects to account for the sorting of teachers to student

populations. As our data show evidence of such sorting (discussed below), we

considered these controls appropriate; to further control for sorting, we also

conducted specification checks including models with school fixed effects.

Results (available upon request) largely replicated those presented here.

We recovered our primary parameters of interest, coefficients representing

the relationship between specific teacher characteristics to student test per-

formance, from ω . To make interpretation of these coefficients easier, we

standardized teacher scores on the mathematics courses, knowledge, and

mind-sets and habits measures across the teacher sample prior to model esti-

mation. We estimated Equation 2 four times for each outcome, testing the

variables first by category and then in an omnibus model. As we had a rela-

tively large number of variables given our sample size, we set a slightly

higher threshold for statistical significance, referring to estimates with p val-

ues between .10 and .05 as marginally significant. For each model estimation,

we conducted a Wald test to examine the joint significance of each category’s

variables in predicting outcomes and also reported the amount of variance in

teacher effects explained by each category (an “adjusted pseudo R-squared”;

see Bacher-Hicks, Chin, Hill & Straiger, 2018).

15.
Hill et al. 1117

Finally, as noted above, prior research has demonstrated imbalances in

teacher qualifications across student populations. To investigate this issue,

we followed Goldhaber et al. (2015) and calculated teacher quality gaps

between advantaged and disadvantaged students (i.e., students who were

Black/Hispanic, FRPL-eligible, ELL, and/or in the bottom quartile of prior-

year state mathematics test performance), and estimated the significance of

these gaps using two-sample t tests. Specifically, we compared the percent-

age of disadvantaged students in our sample taught by teachers that per-

formed poorly, as compared with other teachers in the same district, on our

key measures of quality to the percentage for advantaged students.

Examining Associations Among Teacher Characteristics and

With Student Outcomes

We start by discussing the correlations among teacher characteristics.

Table 2 shows few notable correlations that arose between our independent

Teachers’ reports of completing mathematics content for teachers and

mathematics methods courses correlated strongly (r = .67); correlations

between these variables and college-level mathematics courses were moderate

(r = .44, .48). Consistent with the conventional training and certification pro-

cesses in most states, traditionally certified teachers more often possessed a

bachelor’s degree in education (r = .59). This, along with evidence of multi-

collinearity in our regressions, led us to combine the mathematics content/

methods courses variables and traditional certification/education bachelor’s

degree metrics in our analyses below. Overall, novice teachers in our sample

took fewer courses (mathematics courses, r = –.25; mathematics content

courses, r = –.34; mathematics methods courses, r = –.32), and often did not

possess a master’s degree (r = –.37). These patterns reflect what we would

expect intuitively, as newer teachers have had less time to attain these addi-

tional milestones. Novice teachers also reported feeling less efficacious (r =

Teachers who reported receiving their bachelor’s degree in education were

less likely to also have completed a master’s degree (r = –.31). There was a

notable relationship between reported mathematics courses and effort (i.e.,

time spent grading papers, preparing for class, and tutoring; r = .22), perhaps

a sign that some teachers in our sample had more time or inclination to invest

in their work. Contrary to expectations, teacher completion of mathematics

Finally, as noted above, prior research has demonstrated imbalances in

teacher qualifications across student populations. To investigate this issue,

we followed Goldhaber et al. (2015) and calculated teacher quality gaps

between advantaged and disadvantaged students (i.e., students who were

Black/Hispanic, FRPL-eligible, ELL, and/or in the bottom quartile of prior-

year state mathematics test performance), and estimated the significance of

these gaps using two-sample t tests. Specifically, we compared the percent-

age of disadvantaged students in our sample taught by teachers that per-

formed poorly, as compared with other teachers in the same district, on our

key measures of quality to the percentage for advantaged students.

Examining Associations Among Teacher Characteristics and

With Student Outcomes

We start by discussing the correlations among teacher characteristics.

Table 2 shows few notable correlations that arose between our independent

Teachers’ reports of completing mathematics content for teachers and

mathematics methods courses correlated strongly (r = .67); correlations

between these variables and college-level mathematics courses were moderate

(r = .44, .48). Consistent with the conventional training and certification pro-

cesses in most states, traditionally certified teachers more often possessed a

bachelor’s degree in education (r = .59). This, along with evidence of multi-

collinearity in our regressions, led us to combine the mathematics content/

methods courses variables and traditional certification/education bachelor’s

degree metrics in our analyses below. Overall, novice teachers in our sample

took fewer courses (mathematics courses, r = –.25; mathematics content

courses, r = –.34; mathematics methods courses, r = –.32), and often did not

possess a master’s degree (r = –.37). These patterns reflect what we would

expect intuitively, as newer teachers have had less time to attain these addi-

tional milestones. Novice teachers also reported feeling less efficacious (r =

Teachers who reported receiving their bachelor’s degree in education were

less likely to also have completed a master’s degree (r = –.31). There was a

notable relationship between reported mathematics courses and effort (i.e.,

time spent grading papers, preparing for class, and tutoring; r = .22), perhaps

a sign that some teachers in our sample had more time or inclination to invest

in their work. Contrary to expectations, teacher completion of mathematics

16.
Table 2. Correlations Between Teacher Characteristics.

1 2 3 4 5 6 7 8 9 10 11 12 13

1. Novice 1

2. Math courses −.25 1

3. Math content courses −.34 .48 1

4. Math methods courses −.32 .44 .67 1

5. Traditional certification −.11 −.12 .11 .09 1

6. Education bachelor’s −.16 −.04 .13 .24 .59 1

7. Elementary math certification −.04 .14 .09 .09 −.43 .15 1

8. Master’s −.37 0 −.01 0 .12 −.31 −.12 1

9. MKT/STEL .11 .04 .02 .03 .07 −.05 .01 .04 1

10. Accuracy .1 −.05 −.07 −.07 .1 −.02 −.03 .11 .24 1

11. Efficacy −.22 .1 .01 .05 −.1 0 .11 −.08 .04 −.08 1

12. Locus of control 0 0 .02 .03 .1 .05 .1 .01 0 −.02 −.19 1

13. Effort .01 .22 .14 .14 −.08 .01 −.01 −.12 −.14 −.09 .07 −.2 1

Note. Light gray cells indicate correlations between .30 and .50. Dark gray cells indicate correlations greater than .50. Tetrachoric, polychoric, and

polyserial correlations reported when appropriate. MKT = Mathematical Knowledge for Teaching; STEL = State Test of Education Licensure.

1 2 3 4 5 6 7 8 9 10 11 12 13

1. Novice 1

2. Math courses −.25 1

3. Math content courses −.34 .48 1

4. Math methods courses −.32 .44 .67 1

5. Traditional certification −.11 −.12 .11 .09 1

6. Education bachelor’s −.16 −.04 .13 .24 .59 1

7. Elementary math certification −.04 .14 .09 .09 −.43 .15 1

8. Master’s −.37 0 −.01 0 .12 −.31 −.12 1

9. MKT/STEL .11 .04 .02 .03 .07 −.05 .01 .04 1

10. Accuracy .1 −.05 −.07 −.07 .1 −.02 −.03 .11 .24 1

11. Efficacy −.22 .1 .01 .05 −.1 0 .11 −.08 .04 −.08 1

12. Locus of control 0 0 .02 .03 .1 .05 .1 .01 0 −.02 −.19 1

13. Effort .01 .22 .14 .14 −.08 .01 −.01 −.12 −.14 −.09 .07 −.2 1

Note. Light gray cells indicate correlations between .30 and .50. Dark gray cells indicate correlations greater than .50. Tetrachoric, polychoric, and

polyserial correlations reported when appropriate. MKT = Mathematical Knowledge for Teaching; STEL = State Test of Education Licensure.

17.
Hill et al. 1119

content or methods courses did not strongly associate with our teacher knowl-

edge measure.

Looking further into the measures from the knowledge and mind-sets and

beliefs categories, we observed few additional patterns. As predicted, the two

measures of teachers’ knowledge correlated with one another, though more

weakly than expected (r = .24). Teacher efficacy also negatively but weakly

correlated with locus of control (r = –.19). The former observed relationship

reflects what might be expected intuitively: Our locus-of-control variable

measures the endorsement of a view of fixed intelligence, and teachers appear

to feel more efficacious when they also feel they can influence student learn-

ing. Some relationships were remarkable in their absence. Against expecta-

tions, mathematical content knowledge did not relate to perceived teacher

efficacy; similarly, efficacy only weakly related to teacher accuracy in pre-

dicting student performance on the project test.

Next, we explore how different measures relate to student outcomes.

Table 3 shows regressions predicting student performance on both the state

and project tests. Among self-reported teacher preparation and experiences,

no measures were significant in both the state and project mathematics tests’

final models. This includes teachers’ completion of mathematics content and/

or methods courses, which positively predicted student performance on both

tests, though only the relationship to the project test appeared significant.

Similarly, students taught by novice teachers performed more poorly on both

the state and project tests, corroborating findings of prior research, yet only

the former outcome proved significant. Beyond these variables, no others

within this category significantly related to student outcomes, including the

possession of a master’s degree, the possession of elementary mathematics

certification, and the combined measure for being traditionally certified and

possessing a bachelor’s degree in education. Noticeably, despite the low

overall variance explained by these variables, the Wald test indicated that the

measures together were jointly significant for predicting performance on the

state test, and were marginally significant for the project test.

In the knowledge category, teachers’ MKT/STEL and accuracy scores pre-

dicted student outcomes on both assessments, with the point estimate and

significance for accuracy slightly higher. Despite again observing low

explained variance, the significance of the Wald test examining the joint sig-

nificance of both knowledge measures for both outcomes further supports

theories positing the importance of teacher knowledge for student learning.

Although these associations are not causal, these results suggest that teach-

ing-related mathematical knowledge and predictive accuracy, though corre-

lated with one another, may be individually important, and thus contribute

separately to student growth.

content or methods courses did not strongly associate with our teacher knowl-

edge measure.

Looking further into the measures from the knowledge and mind-sets and

beliefs categories, we observed few additional patterns. As predicted, the two

measures of teachers’ knowledge correlated with one another, though more

weakly than expected (r = .24). Teacher efficacy also negatively but weakly

correlated with locus of control (r = –.19). The former observed relationship

reflects what might be expected intuitively: Our locus-of-control variable

measures the endorsement of a view of fixed intelligence, and teachers appear

to feel more efficacious when they also feel they can influence student learn-

ing. Some relationships were remarkable in their absence. Against expecta-

tions, mathematical content knowledge did not relate to perceived teacher

efficacy; similarly, efficacy only weakly related to teacher accuracy in pre-

dicting student performance on the project test.

Next, we explore how different measures relate to student outcomes.

Table 3 shows regressions predicting student performance on both the state

and project tests. Among self-reported teacher preparation and experiences,

no measures were significant in both the state and project mathematics tests’

final models. This includes teachers’ completion of mathematics content and/

or methods courses, which positively predicted student performance on both

tests, though only the relationship to the project test appeared significant.

Similarly, students taught by novice teachers performed more poorly on both

the state and project tests, corroborating findings of prior research, yet only

the former outcome proved significant. Beyond these variables, no others

within this category significantly related to student outcomes, including the

possession of a master’s degree, the possession of elementary mathematics

certification, and the combined measure for being traditionally certified and

possessing a bachelor’s degree in education. Noticeably, despite the low

overall variance explained by these variables, the Wald test indicated that the

measures together were jointly significant for predicting performance on the

state test, and were marginally significant for the project test.

In the knowledge category, teachers’ MKT/STEL and accuracy scores pre-

dicted student outcomes on both assessments, with the point estimate and

significance for accuracy slightly higher. Despite again observing low

explained variance, the significance of the Wald test examining the joint sig-

nificance of both knowledge measures for both outcomes further supports

theories positing the importance of teacher knowledge for student learning.

Although these associations are not causal, these results suggest that teach-

ing-related mathematical knowledge and predictive accuracy, though corre-

lated with one another, may be individually important, and thus contribute

separately to student growth.

18.
Table 3. Predicting Student Mathematics Test Performance Using Teacher Characteristics.

State test Project test

Prep. and Mind-sets Prep. and Mind-sets

experiences Knowledge and habits All experiences Knowledge and habits All

Novice −0.107* −0.121* −0.022 −0.035

(0.046) (0.045) (0.046) (0.046)

Math courses 0.004 −0.004 0.001 −0.002

(0.014) (0.014) (0.014) (0.014)

Math content/methods 0.012 0.012 0.020** 0.021**

courses (0.008) (0.008) (0.008) (0.007)

Trad. cert./Ed. bachelor’s 0.028 0.027 −0.003 −0.004

(0.020) (0.020) (0.019) (0.019)

El. math cert. −0.004 0.002 −0.045 −0.041

(0.035) (0.034) (0.033) (0.033)

Master’s 0.010 0.008 −0.009 −0.016

(0.031) (0.030) (0.029) (0.029)

MKT/STEL 0.017 0.023† 0.023† 0.023†

(0.013) (0.012) (0.012) (0.012)

Accuracy 0.023† 0.027* 0.030* 0.034**

(0.012) (0.012) (0.012) (0.012)

Efficacy −0.001 −0.001 0.002 0.004

(0.012) (0.012) (0.012) (0.012)

Locus of control 0.004 0.001 −0.004 −0.006

(0.012) (0.012) (0.012) (0.011)

Effort 0.032** 0.034** 0.008 0.005

(0.012) (0.013) (0.012) (0.012)

Variance explained −0.002 0.039 0.048 0.093 0.047 0.062 0.004 0.121

Wald test p value .016 .034 .078 .001 .070 .002 .859 .004

Note. The number of students and teachers in each model is 10,233 and 306, respectively. MKT = Mathematical Knowledge for Teaching; STEL = State Test of Education

Licensure.

†p < .10. *p < .05. **p < .01. ***p < .001.

State test Project test

Prep. and Mind-sets Prep. and Mind-sets

experiences Knowledge and habits All experiences Knowledge and habits All

Novice −0.107* −0.121* −0.022 −0.035

(0.046) (0.045) (0.046) (0.046)

Math courses 0.004 −0.004 0.001 −0.002

(0.014) (0.014) (0.014) (0.014)

Math content/methods 0.012 0.012 0.020** 0.021**

courses (0.008) (0.008) (0.008) (0.007)

Trad. cert./Ed. bachelor’s 0.028 0.027 −0.003 −0.004

(0.020) (0.020) (0.019) (0.019)

El. math cert. −0.004 0.002 −0.045 −0.041

(0.035) (0.034) (0.033) (0.033)

Master’s 0.010 0.008 −0.009 −0.016

(0.031) (0.030) (0.029) (0.029)

MKT/STEL 0.017 0.023† 0.023† 0.023†

(0.013) (0.012) (0.012) (0.012)

Accuracy 0.023† 0.027* 0.030* 0.034**

(0.012) (0.012) (0.012) (0.012)

Efficacy −0.001 −0.001 0.002 0.004

(0.012) (0.012) (0.012) (0.012)

Locus of control 0.004 0.001 −0.004 −0.006

(0.012) (0.012) (0.012) (0.011)

Effort 0.032** 0.034** 0.008 0.005

(0.012) (0.013) (0.012) (0.012)

Variance explained −0.002 0.039 0.048 0.093 0.047 0.062 0.004 0.121

Wald test p value .016 .034 .078 .001 .070 .002 .859 .004

Note. The number of students and teachers in each model is 10,233 and 306, respectively. MKT = Mathematical Knowledge for Teaching; STEL = State Test of Education

Licensure.

†p < .10. *p < .05. **p < .01. ***p < .001.

19.
Hill et al. 1121

In the mind-sets and habits category, neither the efficacy nor the locus-

of-control measures predicted performance on the state or project assess-

ment in the final analysis. By contrast, teachers’ self-reported effort—the

number of hours spent grading mathematics homework, preparing mathe-

matics lessons, and tutoring students outside of regular school hours—pre-

dicted performance on the state but not on the project test. To check the

intuition that tutoring may have driven this result, perhaps as teachers

helped students prepare for state assessments, we removed the tutoring item

from the scale and found the same result (b = .035, p < .01). Despite this

striking finding, this category explained again only a small (5%) amount of

variance in teacher effects on student state test performance and the Wald

test was just marginally significant.

To determine the extent to which the significant relationships between our

measures of teacher characteristics and student test scores is meaningful, we

make two comparisons. First, we compare the coefficient sizes in Table 3 with

the size of the teacher-level SD in student scores on the state tests (0.16) and

project-developed tests (0.14). Second, we compare the relationship between

teacher characteristics and test scores with those of key student characteristics

and test scores, such as FRPL-eligibility ( βState = −.05; β Project − developed = −.04).

Through these comparisons, we see that the coefficient on state test perfor-

mance for a novice teacher ( β = −.12) is sizable; students taught by a more

veteran teacher in our sample had test score gains of nearly three quarters of

those taught by a teacher 1-SD above average for raising student state test

scores; this coefficient also completely closes the gain-gap for students who

are FRPL-eligible. On the contrary, being taught by a teacher 1-SD above

average on other significant teacher predictors such as MKT/STEL, accuracy,

and self-reported effort does not yield commensurate gains. In each case, stu-

dents taught by an above-average teacher gain less than one fifth of that

observed for students taught by teacher 1-SD above average for raising student

test scores. However, teacher performance on these measures still accounts for

a significant proportion of the gain-gap associated with FRPL-eligibility.

Finally, as noted above, we assessed the consistency of relationships

across the two different student tests (i.e., criterion stability). Findings here

perhaps shed light on the mostly mixed results from prior studies of the edu-

cation production function. Conflicting results, for instance, could have been

caused by studies occurring in states with different teacher education and

certification pathways (some more effective than others), or, as here, by dif-

ferences in the sensitivity of the tests used to measure teacher characteristics.

Our results suggest conflicting findings for three variables—teacher experi-

ence, enrollment in mathematics content and/or methods courses, and effort.

Notably, however, nonsignificant findings were similarly consistent. Thus,

In the mind-sets and habits category, neither the efficacy nor the locus-

of-control measures predicted performance on the state or project assess-

ment in the final analysis. By contrast, teachers’ self-reported effort—the

number of hours spent grading mathematics homework, preparing mathe-

matics lessons, and tutoring students outside of regular school hours—pre-

dicted performance on the state but not on the project test. To check the

intuition that tutoring may have driven this result, perhaps as teachers

helped students prepare for state assessments, we removed the tutoring item

from the scale and found the same result (b = .035, p < .01). Despite this

striking finding, this category explained again only a small (5%) amount of

variance in teacher effects on student state test performance and the Wald

test was just marginally significant.

To determine the extent to which the significant relationships between our

measures of teacher characteristics and student test scores is meaningful, we

make two comparisons. First, we compare the coefficient sizes in Table 3 with

the size of the teacher-level SD in student scores on the state tests (0.16) and

project-developed tests (0.14). Second, we compare the relationship between

teacher characteristics and test scores with those of key student characteristics

and test scores, such as FRPL-eligibility ( βState = −.05; β Project − developed = −.04).

Through these comparisons, we see that the coefficient on state test perfor-

mance for a novice teacher ( β = −.12) is sizable; students taught by a more

veteran teacher in our sample had test score gains of nearly three quarters of

those taught by a teacher 1-SD above average for raising student state test

scores; this coefficient also completely closes the gain-gap for students who

are FRPL-eligible. On the contrary, being taught by a teacher 1-SD above

average on other significant teacher predictors such as MKT/STEL, accuracy,

and self-reported effort does not yield commensurate gains. In each case, stu-

dents taught by an above-average teacher gain less than one fifth of that

observed for students taught by teacher 1-SD above average for raising student

test scores. However, teacher performance on these measures still accounts for

a significant proportion of the gain-gap associated with FRPL-eligibility.

Finally, as noted above, we assessed the consistency of relationships

across the two different student tests (i.e., criterion stability). Findings here

perhaps shed light on the mostly mixed results from prior studies of the edu-

cation production function. Conflicting results, for instance, could have been

caused by studies occurring in states with different teacher education and

certification pathways (some more effective than others), or, as here, by dif-

ferences in the sensitivity of the tests used to measure teacher characteristics.

Our results suggest conflicting findings for three variables—teacher experi-

ence, enrollment in mathematics content and/or methods courses, and effort.

Notably, however, nonsignificant findings were similarly consistent. Thus,

20.
1122 Educational Policy 33(7)

we found more consistency across tests within the same sample than sug-

gested by the aggregate findings from prior literature.

Investigating the Distribution of Teacher Characteristics Across

Student Populations

To examine the distribution of key teacher characteristics across student popu-

lations within districts (our second research question), Table 4 displays the

exposure of disadvantaged students to teacher characteristics associated with

poorer performance, as judged by our models above. Black and Hispanic stu-

dents, FRPL-eligible students, ELL-students, and low-performing students

were all exposed to novice teachers and teachers performing in the bottom

quartile of their districts on the MKT/STEL and accuracy measures more fre-

quently than their more advantaged counterparts. These findings were unsur-

prising and matched intuition and prior research (Hill & Lubienski, 2007;

Loeb & Reininger, 2004). Conversely, we found the opposite pattern for

teacher effort; disadvantaged students were less frequently exposed to teach-

ers in the bottom quartile of their district on this measure. These findings sug-

gest that teachers may adjust their behaviors in response to the needs of

students they teach. Interestingly, exposure to teachers with less mathematics

content and/or methods coursework was generally evenly distributed across

groups of students, with the exception of the exposure gap between FRPL-

versus non–FRPL-eligible students. Mathematics content and/or methods

courses may be prescribed by local or state guidelines, or incentivized by dis-

tricts themselves, resulting in a relatively even distribution across student

populations. Overall, however, the results depicted in Table 4 suggest that the

teacher-level characteristics we identified in earlier analyses as important for

student learning varied inequitably across among students in our sample. We

return to these results and their implications for policy in our “Discussion”

Finally, we conducted a number of checks of our results, looking for inter-

action effects between key variables (e.g., whether effort yielded differential

benefits by teacher knowledge) and between key variables and student popu-

lations (e.g., whether there were consistent patterns in the association between

resources within student subgroups). We found no significant interaction

We initiated our work by noting that although policy makers may wish for

clear guidance regarding characteristics of effective teachers, scholarship on

we found more consistency across tests within the same sample than sug-

gested by the aggregate findings from prior literature.

Investigating the Distribution of Teacher Characteristics Across

Student Populations

To examine the distribution of key teacher characteristics across student popu-

lations within districts (our second research question), Table 4 displays the

exposure of disadvantaged students to teacher characteristics associated with

poorer performance, as judged by our models above. Black and Hispanic stu-

dents, FRPL-eligible students, ELL-students, and low-performing students

were all exposed to novice teachers and teachers performing in the bottom

quartile of their districts on the MKT/STEL and accuracy measures more fre-

quently than their more advantaged counterparts. These findings were unsur-

prising and matched intuition and prior research (Hill & Lubienski, 2007;

Loeb & Reininger, 2004). Conversely, we found the opposite pattern for

teacher effort; disadvantaged students were less frequently exposed to teach-

ers in the bottom quartile of their district on this measure. These findings sug-

gest that teachers may adjust their behaviors in response to the needs of

students they teach. Interestingly, exposure to teachers with less mathematics

content and/or methods coursework was generally evenly distributed across

groups of students, with the exception of the exposure gap between FRPL-

versus non–FRPL-eligible students. Mathematics content and/or methods

courses may be prescribed by local or state guidelines, or incentivized by dis-

tricts themselves, resulting in a relatively even distribution across student

populations. Overall, however, the results depicted in Table 4 suggest that the

teacher-level characteristics we identified in earlier analyses as important for

student learning varied inequitably across among students in our sample. We

return to these results and their implications for policy in our “Discussion”

Finally, we conducted a number of checks of our results, looking for inter-

action effects between key variables (e.g., whether effort yielded differential

benefits by teacher knowledge) and between key variables and student popu-

lations (e.g., whether there were consistent patterns in the association between

resources within student subgroups). We found no significant interaction

We initiated our work by noting that although policy makers may wish for

clear guidance regarding characteristics of effective teachers, scholarship on

21.
Table 4. Exposure Rates to Low-Performing Teachers on Key Teacher Characteristics.

Race/ethnicity FRPL-eligibility

Black/ Non–Black/

Panel A Hispanic Hispanic Difference FRPL Non-FRPL Difference

Novice Teacher 0.07 0.03 0.04*** 0.07 0.03 0.04***

Low Math Content/Methods Courses Teacher 0.45 0.42 0.03 0.41 0. 45 −0.04***

Low MKT/STEL Teacher 0.23 0.16 0.07*** 0.23 0.17 0.06***

Low Accuracy Teacher 0.22 0.20 0.02* 0.22 0.19 0.03**

Low Effort Teacher 0.21 0.26 −0.05*** 0.21 0.26 −0.05***

Quartile of prior state test

ELL status performance

Non-

Panel B ELL Non-ELL Difference Lowest lowest Difference

Novice Teacher 0.09 0.05 0.04*** 0.07 0.05 0.02***

Low Math Content/Methods Courses Teacher 0.43 0.43 0.00 0.44 0.42 0.02

Low MKT/STEL Teacher 0.27 0.19 0.08*** 0.25 0.19 0.06***

Low Accuracy Teacher 0.25 0.20 0.05*** 0.24 0.20 0.04***

Low Effort Teacher 0.19 0.24 −0.05*** 0.19 0.24 −0.05***

Note. FRPL = free or reduced-price lunch; MKT = Mathematical Knowledge for Teaching; STEL = State Test of Education Licensure; ELL = English

language learner.

†p < .10. *p < .05. **p < .01. ***p < .001 (two-sample t test).

Race/ethnicity FRPL-eligibility

Black/ Non–Black/

Panel A Hispanic Hispanic Difference FRPL Non-FRPL Difference

Novice Teacher 0.07 0.03 0.04*** 0.07 0.03 0.04***

Low Math Content/Methods Courses Teacher 0.45 0.42 0.03 0.41 0. 45 −0.04***

Low MKT/STEL Teacher 0.23 0.16 0.07*** 0.23 0.17 0.06***

Low Accuracy Teacher 0.22 0.20 0.02* 0.22 0.19 0.03**

Low Effort Teacher 0.21 0.26 −0.05*** 0.21 0.26 −0.05***

Quartile of prior state test

ELL status performance

Non-

Panel B ELL Non-ELL Difference Lowest lowest Difference

Novice Teacher 0.09 0.05 0.04*** 0.07 0.05 0.02***

Low Math Content/Methods Courses Teacher 0.43 0.43 0.00 0.44 0.42 0.02

Low MKT/STEL Teacher 0.27 0.19 0.08*** 0.25 0.19 0.06***

Low Accuracy Teacher 0.25 0.20 0.05*** 0.24 0.20 0.04***

Low Effort Teacher 0.19 0.24 −0.05*** 0.19 0.24 −0.05***

Note. FRPL = free or reduced-price lunch; MKT = Mathematical Knowledge for Teaching; STEL = State Test of Education Licensure; ELL = English

language learner.

†p < .10. *p < .05. **p < .01. ***p < .001 (two-sample t test).

22.
1124 Educational Policy 33(7)

this topic has returned mixed results and, in many cases, studies that exam-

ine only a handful of characteristics in isolation (for exceptions, see, for

example, Boonen et al., 2014; Campbell et al., 2014; Grubb, 2008; Palardy

& Rumberger, 2008). By bringing together characteristics from three main

categories—teacher preparation and experience, knowledge, and mind-sets

and habits—we assessed their joint associations with student learning. We

also explored the criterion consistency of our findings, as we considered

student learning as measured on a state standardized assessment and on a

project mathematics assessment. In doing so, our study not only comple-

ments but also extends existing approaches examining the contribution of

teacher-level characteristics to student learning in two significant ways.

First, it explores a noticeably more comprehensive list of teacher attributes

compared with those considered in prior studies; second, we examine the

distribution of key teacher characteristics across student populations within

districts, building on the work of Choi (2010), Goldhaber et al. (2015), and

Schultz (2014). Finally, our work also allowed us to examine correlations

between teacher characteristics.

We found most correlations in our data to be mild, at .20 or low. Variables

that represent teacher preparation and experiences proved to be one excep-

tion, with factors such as coursework, an education major, certification, expe-

rience, and a master’s degree correlating at .30 or higher. This may shed light

on the mixed evidence for many of these variables in the extant economics of

education literature, where omitted variable bias may lead to disparate results.

These correlational analyses also revealed that expected relationships

between such variables do not always materialize. This was particularly

interesting in the case of mathematics coursework, teacher knowledge, and

efficacy, where we expected to see strong relationships.

By and large, results from both of the student-level outcomes under con-

sideration pointed to the same characteristics as potentially important.

Consistent with some prior literature (Monk, 1994; Rice, 2003) and the atten-

tion paid to such courses in educational systems worldwide (Tatto et al.,

2008), we found that the completion of mathematics content and/or methods

courses positively related with student learning on both outcomes, with an

observed significant relationship for the project test. This is remarkable in an

era in which many teacher preparation programs—particularly alternative

entry pathways—do not feature content-specific teaching coursework. It sug-

gests that such coursework may be an important support for elementary

teachers (Sleeter, 2014); however, the possibility that selection effects (e.g.,

teachers more comfortable with mathematics may enroll in more such

courses) influenced our findings cannot be ruled out. Because this work took

place in only four districts, we also cannot be sure whether the associations

this topic has returned mixed results and, in many cases, studies that exam-

ine only a handful of characteristics in isolation (for exceptions, see, for

example, Boonen et al., 2014; Campbell et al., 2014; Grubb, 2008; Palardy

& Rumberger, 2008). By bringing together characteristics from three main

categories—teacher preparation and experience, knowledge, and mind-sets

and habits—we assessed their joint associations with student learning. We

also explored the criterion consistency of our findings, as we considered

student learning as measured on a state standardized assessment and on a

project mathematics assessment. In doing so, our study not only comple-

ments but also extends existing approaches examining the contribution of

teacher-level characteristics to student learning in two significant ways.

First, it explores a noticeably more comprehensive list of teacher attributes

compared with those considered in prior studies; second, we examine the

distribution of key teacher characteristics across student populations within

districts, building on the work of Choi (2010), Goldhaber et al. (2015), and

Schultz (2014). Finally, our work also allowed us to examine correlations

between teacher characteristics.

We found most correlations in our data to be mild, at .20 or low. Variables

that represent teacher preparation and experiences proved to be one excep-

tion, with factors such as coursework, an education major, certification, expe-

rience, and a master’s degree correlating at .30 or higher. This may shed light

on the mixed evidence for many of these variables in the extant economics of

education literature, where omitted variable bias may lead to disparate results.

These correlational analyses also revealed that expected relationships

between such variables do not always materialize. This was particularly

interesting in the case of mathematics coursework, teacher knowledge, and

efficacy, where we expected to see strong relationships.

By and large, results from both of the student-level outcomes under con-

sideration pointed to the same characteristics as potentially important.

Consistent with some prior literature (Monk, 1994; Rice, 2003) and the atten-

tion paid to such courses in educational systems worldwide (Tatto et al.,

2008), we found that the completion of mathematics content and/or methods

courses positively related with student learning on both outcomes, with an

observed significant relationship for the project test. This is remarkable in an

era in which many teacher preparation programs—particularly alternative

entry pathways—do not feature content-specific teaching coursework. It sug-

gests that such coursework may be an important support for elementary

teachers (Sleeter, 2014); however, the possibility that selection effects (e.g.,

teachers more comfortable with mathematics may enroll in more such

courses) influenced our findings cannot be ruled out. Because this work took

place in only four districts, we also cannot be sure whether the associations

23.
Hill et al. 1125

we observe are specific to the teacher education programs that serve these

districts, or whether this association between coursework and student out-

comes holds generally.

Similarly, in line with recent research findings on teacher knowledge, the

two teacher mathematics knowledge measures we employed—teacher accu-

racy in predicting student mathematics test performance and MKT/STEL—

positively related to student outcomes. This supports the importance of

teacher knowledge of content and its teaching and of what students know and

do not know—both components of Shulman’s conceptualization of teacher

knowledge and Ball, Thames, and Phelps’s (2008) notion of MKT. The mod-

els in this article improve upon those offered in most prior research, in that

they are well controlled for related teacher characteristics and knowledge,

suggesting that these associations were not driven by omitted correlates such

as efficacy and mathematics coursework. Interestingly, the variables repre-

senting mathematics methods/content courses and teacher knowledge did not

relate to one another, suggesting that each had an independent pathway

through which they related to student outcomes.

As in other reports (e.g., Kane et al., 2008), lack of teaching experience

related negatively to student outcomes. Here, however, the measure was only

significant for the state test. One intriguing possibility is that the association

between novice teachers and student outcomes may result as much from

familiarity with the state standardized test as it does from lagging effective-

ness in the classroom. Novice teachers may not optimally adjust their cur-

riculum and pacing to align with the state assessment; they also may be

unfamiliar with question formats and topics. A similar study found no rela-

tionship between experience and state test scores, with a positive effect

observed between novice teachers and an alternative test in mathematics

(Kane & Staiger, 2012). Thus, this is an issue for further analyses, potentially

via a review of available evidence on this topic.

Our findings also suggest attention to teacher effort, measured as invest-

ment in noninstructional work hours, which here had a positive association

with student learning as measured by state test results. Interestingly, this posi-

tive association did not appear to be driven by tutoring alone. If students

benefit when their teacher spends more time grading papers and preparing for

class, then arranging for a greater ratio of noninstructional to instructional

work hours in U.S. schools—and cultivating knowledge about the productive

use of that time—becomes imperative. However, because we cannot make a

causal attribution in this study, this issue warrants further investigation.

Several teacher characteristics thought to predict increased student perfor-

mance in mathematics did not do so in this sample. Teacher self-efficacy was

one such characteristic. Although this is a frequently studied teacher belief,

we observe are specific to the teacher education programs that serve these

districts, or whether this association between coursework and student out-

comes holds generally.

Similarly, in line with recent research findings on teacher knowledge, the

two teacher mathematics knowledge measures we employed—teacher accu-

racy in predicting student mathematics test performance and MKT/STEL—

positively related to student outcomes. This supports the importance of

teacher knowledge of content and its teaching and of what students know and

do not know—both components of Shulman’s conceptualization of teacher

knowledge and Ball, Thames, and Phelps’s (2008) notion of MKT. The mod-

els in this article improve upon those offered in most prior research, in that

they are well controlled for related teacher characteristics and knowledge,

suggesting that these associations were not driven by omitted correlates such

as efficacy and mathematics coursework. Interestingly, the variables repre-

senting mathematics methods/content courses and teacher knowledge did not

relate to one another, suggesting that each had an independent pathway

through which they related to student outcomes.

As in other reports (e.g., Kane et al., 2008), lack of teaching experience

related negatively to student outcomes. Here, however, the measure was only

significant for the state test. One intriguing possibility is that the association

between novice teachers and student outcomes may result as much from

familiarity with the state standardized test as it does from lagging effective-

ness in the classroom. Novice teachers may not optimally adjust their cur-

riculum and pacing to align with the state assessment; they also may be

unfamiliar with question formats and topics. A similar study found no rela-

tionship between experience and state test scores, with a positive effect

observed between novice teachers and an alternative test in mathematics

(Kane & Staiger, 2012). Thus, this is an issue for further analyses, potentially

via a review of available evidence on this topic.

Our findings also suggest attention to teacher effort, measured as invest-

ment in noninstructional work hours, which here had a positive association

with student learning as measured by state test results. Interestingly, this posi-

tive association did not appear to be driven by tutoring alone. If students

benefit when their teacher spends more time grading papers and preparing for

class, then arranging for a greater ratio of noninstructional to instructional

work hours in U.S. schools—and cultivating knowledge about the productive

use of that time—becomes imperative. However, because we cannot make a

causal attribution in this study, this issue warrants further investigation.

Several teacher characteristics thought to predict increased student perfor-

mance in mathematics did not do so in this sample. Teacher self-efficacy was

one such characteristic. Although this is a frequently studied teacher belief,

24.
1126 Educational Policy 33(7)

and even though we used a widely disseminated metric, it neither correlated

strongly with hypothetically related constructs (e.g., teacher knowledge,

locus of control) nor appeared related to student outcomes. We also note,

parenthetically, that the version of the self-efficacy instrument used here

seemed to us more a self-report of teaching expertise than efficacy as envi-

sioned by theorists such as Bandura (1997), for whom the construct also

included aspects of grit and task persistence. Locus of control also saw rela-

tionships close to zero.

We found imbalances of key teacher characteristics across populations of

students. Specifically, students from disadvantaged groups more frequently

had novice teachers and those with lower knowledge scores. These findings,

which hearken to sociologist Robert Merton’s (1968) notion of accumulated

advantage—”the rich get richer and the poor get poorer”—align with other

related findings and scholarly discussions both in the United States (Darling-

Hammond, 2010) and worldwide (Schleicher & Organization for Economic

Co-Operation and Development, 2014). This research implies that the stu-

dents in most need of high-quality teachers are not afforded them. Although

this imbalance—a significant problem—cannot explain the entire achieve-

ment gap between privileged and less privileged students, it undoubtedly

explains some. Advocates for better hiring and placement practices (e.g.,

Rutledge, Harris, Thompson, & Ingle, 2008) are correct in noting that this is

a solvable problem (Liu & Johnson, 2006), and in fact, the Every Student

Succeeds Act (ESSA) requires that states define ineffective teachers and

ensure that poor and minority students are not taught disproportionately by

such teachers. In mathematics, metrics such as teacher certification test

scores in mathematics (related, potentially, to MKT), teachers’ content/meth-

ods coursework, and years of experience could prove relatively inexpensive

ways for states to collect information about inequities in distribution of the

teaching workforce, and to incent districts to act upon such inequities by

publicizing that information while also supporting districts with poor-quality

teachers make improvements in their hiring process.

In larger perspective, these findings suggest that despite some consistent

patterns, there does not seem to be one teacher characteristic that exhibited a

strong relationship to teacher effectiveness in mathematics. Even variables

found to significantly predict student mathematics outcomes had small coef-

ficients. In addition, our variables explained a modest, at best, percentage of

the variance in student learning, even in conjunction with one another. This

result mirrors outcomes from similar studies, including those that examine

teacher characteristics and those that measure instructional quality via obser-

vation rubrics (Kane & Staiger, 2012; Stronge, Ward, & Grant, 2011). One

perspective on these findings is that there remain unexamined teacher-level

and even though we used a widely disseminated metric, it neither correlated

strongly with hypothetically related constructs (e.g., teacher knowledge,

locus of control) nor appeared related to student outcomes. We also note,

parenthetically, that the version of the self-efficacy instrument used here

seemed to us more a self-report of teaching expertise than efficacy as envi-

sioned by theorists such as Bandura (1997), for whom the construct also

included aspects of grit and task persistence. Locus of control also saw rela-

tionships close to zero.

We found imbalances of key teacher characteristics across populations of

students. Specifically, students from disadvantaged groups more frequently

had novice teachers and those with lower knowledge scores. These findings,

which hearken to sociologist Robert Merton’s (1968) notion of accumulated

advantage—”the rich get richer and the poor get poorer”—align with other

related findings and scholarly discussions both in the United States (Darling-

Hammond, 2010) and worldwide (Schleicher & Organization for Economic

Co-Operation and Development, 2014). This research implies that the stu-

dents in most need of high-quality teachers are not afforded them. Although

this imbalance—a significant problem—cannot explain the entire achieve-

ment gap between privileged and less privileged students, it undoubtedly

explains some. Advocates for better hiring and placement practices (e.g.,

Rutledge, Harris, Thompson, & Ingle, 2008) are correct in noting that this is

a solvable problem (Liu & Johnson, 2006), and in fact, the Every Student

Succeeds Act (ESSA) requires that states define ineffective teachers and

ensure that poor and minority students are not taught disproportionately by

such teachers. In mathematics, metrics such as teacher certification test

scores in mathematics (related, potentially, to MKT), teachers’ content/meth-

ods coursework, and years of experience could prove relatively inexpensive

ways for states to collect information about inequities in distribution of the

teaching workforce, and to incent districts to act upon such inequities by

publicizing that information while also supporting districts with poor-quality

teachers make improvements in their hiring process.

In larger perspective, these findings suggest that despite some consistent

patterns, there does not seem to be one teacher characteristic that exhibited a

strong relationship to teacher effectiveness in mathematics. Even variables

found to significantly predict student mathematics outcomes had small coef-

ficients. In addition, our variables explained a modest, at best, percentage of

the variance in student learning, even in conjunction with one another. This

result mirrors outcomes from similar studies, including those that examine

teacher characteristics and those that measure instructional quality via obser-

vation rubrics (Kane & Staiger, 2012; Stronge, Ward, & Grant, 2011). One

perspective on these findings is that there remain unexamined teacher-level

25.
Hill et al. 1127

variables that help explain student outcomes; another is that factors beyond

the scope of this study, including classroom climate and enacted instructional

practice, explain student outcomes; a third is that given the complexity of the

education production function and noise present in student test score data, no

combination of measured factors will result in a model with strong predictive

Because this study is not experimental, we cannot rule out the possibility

that the above findings (and nonfindings) are driven by selection effects.

Individuals with more aptitude for teaching may invest in more content and

methods coursework, and teachers with more mathematical knowledge may

be hired into more affluent schools serving more academically advanced stu-

dents. However, our models include many controls at the teacher, student,

and school levels, and recent work in the economics of education has sug-

gested that including prior student achievement in such models serves as an

adequate control for teacher sorting (Chetty et al., 2014). This suggests that

while not causal, our findings can help formulate strong hypotheses for future

Our findings can certainly be useful, even descriptively, to LEAs inter-

ested in hiring, retaining, and remunerating the most qualified candidates.

One suggestion would be to screen for easily observable variables, including

teachers’ mathematical knowledge and background in mathematics-specific

coursework, and teaching experience. LEAs may also search for proxies for

teacher effort, such as existing metrics of conscientiousness (e.g., McIlveen

& Perera, 2016). By contrast, LEAs should not hire based on certification,

certification specialization in mathematics, or advanced degrees. The mas-

ter’s degree finding also suggests that LEAs may wish to rethink automatic

salary increases that often accompany the acquisition of advanced degrees

(see also Roza & Miller, 2009).

Similarly, the two aspects of teacher knowledge found to contribute to

student outcomes—mathematical knowledge and accuracy in predicting stu-

dent outcomes—appear amenable to improvement through professional

development programs, particularly those focused on mathematics content

and formative assessment (e.g., see Lang, Schoen, LaVenia, & Oberlin,

2014). Although more research is needed to further validate these relation-

ships, the results of our study, along with those of qualitative studies docu-

menting positive associations between teachers’ knowledge of student

thinking and instructional quality (e.g., Bray, 2011; Even & Tirosh, 2008),

suggest that LEAs ought to support teachers’ development and deepen their

respective knowledge in these domains.

Finally, this study provides guidance for LEAs interested in ensuring that

teacher expertise is distributed equitably over student populations. LEAs

variables that help explain student outcomes; another is that factors beyond

the scope of this study, including classroom climate and enacted instructional

practice, explain student outcomes; a third is that given the complexity of the

education production function and noise present in student test score data, no

combination of measured factors will result in a model with strong predictive

Because this study is not experimental, we cannot rule out the possibility

that the above findings (and nonfindings) are driven by selection effects.

Individuals with more aptitude for teaching may invest in more content and

methods coursework, and teachers with more mathematical knowledge may

be hired into more affluent schools serving more academically advanced stu-

dents. However, our models include many controls at the teacher, student,

and school levels, and recent work in the economics of education has sug-

gested that including prior student achievement in such models serves as an

adequate control for teacher sorting (Chetty et al., 2014). This suggests that

while not causal, our findings can help formulate strong hypotheses for future

Our findings can certainly be useful, even descriptively, to LEAs inter-

ested in hiring, retaining, and remunerating the most qualified candidates.

One suggestion would be to screen for easily observable variables, including

teachers’ mathematical knowledge and background in mathematics-specific

coursework, and teaching experience. LEAs may also search for proxies for

teacher effort, such as existing metrics of conscientiousness (e.g., McIlveen

& Perera, 2016). By contrast, LEAs should not hire based on certification,

certification specialization in mathematics, or advanced degrees. The mas-

ter’s degree finding also suggests that LEAs may wish to rethink automatic

salary increases that often accompany the acquisition of advanced degrees

(see also Roza & Miller, 2009).

Similarly, the two aspects of teacher knowledge found to contribute to

student outcomes—mathematical knowledge and accuracy in predicting stu-

dent outcomes—appear amenable to improvement through professional

development programs, particularly those focused on mathematics content

and formative assessment (e.g., see Lang, Schoen, LaVenia, & Oberlin,

2014). Although more research is needed to further validate these relation-

ships, the results of our study, along with those of qualitative studies docu-

menting positive associations between teachers’ knowledge of student

thinking and instructional quality (e.g., Bray, 2011; Even & Tirosh, 2008),

suggest that LEAs ought to support teachers’ development and deepen their

respective knowledge in these domains.

Finally, this study provides guidance for LEAs interested in ensuring that

teacher expertise is distributed equitably over student populations. LEAs

26.
1128 Educational Policy 33(7)

may make use of data collected at the point of hire (e.g., course transcripts,

content-specific certification test scores) or in administrative data (e.g., expe-

rience) to understand the distribution of teacher characteristics across student

populations. LEAs may also wish to collect additional data on teachers’ accu-

racy in assessing student understanding, and around teacher effort. This

advice applies to LEAs engaged in meeting ESSA reporting requirements

regarding teacher quality and student populations.

The lack of a single “silver-bullet” teacher characteristic predicting stu-

dent outcomes also contains lessons for research, namely that future research

studies of this type should contain as many measures as is practically feasi-

ble. Without extensive coverage of key teacher traits, models may suffer from

omitted variable bias. The results of past research, which show conflicting

evidence regarding key variables such as mathematics methods and content

courses and teacher efficacy, also indicate that replication research of this sort

is warranted, ideally with larger and more representative datasets. Scholars

may also wish to design studies that capture variability in key variables—for

example, to discern the relative effectiveness of specific forms of mathemat-

ics-related teacher preparation coursework (see, for example, Boyd,

Grossman, Lankford, Loeb, & Wyckoff, 2009) and specific forms of teacher

knowledge and teaching experience. With such research, we could sharpen

our lessons and suggestions for practitioners and policy makers.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research,

authorship, and/or publication of this article.

The author(s) disclosed receipt of the following financial support for the research,

authorship, and/or publication of this article: The research was supported in part by

Grant R305C090023 from the Institute of Education Sciences.

1. Intraclass correlations were adjusted for the modal number of accuracy items

teachers responded to. This adjustment provided an estimate of reliability that

was more reflective of measure scores, which incorporated teacher responses to

several items as opposed to a single item.

2. Because we asked teachers to answer questions regarding locus of control only

in 2011-2012, we did not use the model from Equation 1 to estimate scores for

teachers on this measure.

3. Teacher accuracy scores were included in the model at the teacher-grade level, and

the indicator for being a novice teacher was considered at the teacher–year level.

may make use of data collected at the point of hire (e.g., course transcripts,

content-specific certification test scores) or in administrative data (e.g., expe-

rience) to understand the distribution of teacher characteristics across student

populations. LEAs may also wish to collect additional data on teachers’ accu-

racy in assessing student understanding, and around teacher effort. This

advice applies to LEAs engaged in meeting ESSA reporting requirements

regarding teacher quality and student populations.

The lack of a single “silver-bullet” teacher characteristic predicting stu-

dent outcomes also contains lessons for research, namely that future research

studies of this type should contain as many measures as is practically feasi-

ble. Without extensive coverage of key teacher traits, models may suffer from

omitted variable bias. The results of past research, which show conflicting

evidence regarding key variables such as mathematics methods and content

courses and teacher efficacy, also indicate that replication research of this sort

is warranted, ideally with larger and more representative datasets. Scholars

may also wish to design studies that capture variability in key variables—for

example, to discern the relative effectiveness of specific forms of mathemat-

ics-related teacher preparation coursework (see, for example, Boyd,

Grossman, Lankford, Loeb, & Wyckoff, 2009) and specific forms of teacher

knowledge and teaching experience. With such research, we could sharpen

our lessons and suggestions for practitioners and policy makers.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research,

authorship, and/or publication of this article.

The author(s) disclosed receipt of the following financial support for the research,

authorship, and/or publication of this article: The research was supported in part by

Grant R305C090023 from the Institute of Education Sciences.

1. Intraclass correlations were adjusted for the modal number of accuracy items

teachers responded to. This adjustment provided an estimate of reliability that

was more reflective of measure scores, which incorporated teacher responses to

several items as opposed to a single item.

2. Because we asked teachers to answer questions regarding locus of control only

in 2011-2012, we did not use the model from Equation 1 to estimate scores for

teachers on this measure.

3. Teacher accuracy scores were included in the model at the teacher-grade level, and

the indicator for being a novice teacher was considered at the teacher–year level.

27.
Hill et al. 1129

Aaronson, D., Barrow, L., & Sander, W. (2007). Teachers and student achievement

in the Chicago public high schools. Journal of Labor Economics, 25, 95-135.

Anderson, R. N., Greene, M. L., & Loewen, P. S. (1988). Relationships among teach-

ers’ and students’ thinking skills, sense of efficacy, and student achievement.

Alberta Journal of Educational Research, 34, 148-165.

Bacher-Hicks, A., Chin, M. J., Hill, H. C., & Staiger, D. O. (2018). Explaining

teacher effects on achievement using commonly found teacher-level predictors.

Manuscript submitted for publication.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching:

What makes it special? Journal of Teacher Education, 59, 389-407.

Bandura, A. (1997). Self-efficacy: The exercise of control. New York, NY: W. H. Freeman.

Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A., . . . Tsai,

Y. M. (2010). Teachers’ mathematical knowledge, cognitive activation in the

classroom, and student progress. American Educational Research Journal, 47,

133-180.

Begle, E. G. (1979). Critical variables in mathematics education: Findings from a

survey of the empirical literature. Washington, DC: Mathematical Association of

America and the Council of Teachers of Mathematics.

Bell, C. A., Wilson, S. M., Higgins, T., & McCoach, D. B. (2010). Measuring the

effects of professional development on teacher knowledge: The case of devel-

oping mathematical ideas. Journal for Research in Mathematics Education, 41,

479-512.

Berman, P., & McLaughlin, M. W. (1977). Federal programs supporting educational

change, Volume VII: Factors affecting implementation and continuation. Santa

Monica, CA: The RAND Corporation.

Boonen, T., Van Damme, J., & Onghena, P. (2014). Teacher effects on student

achievement in first grade: Which aspects matter most? School Effectiveness and

School Improvement, 25, 126-152.

Boyd, D. J., Grossman, P. L., Lankford, H., Loeb, S., & Wyckoff, J. (2009). Teacher

preparation and student achievement. Educational Evaluation and Policy

Analysis, 31, 416-440.

Bray, W. S. (2011). A collective case study of the influence of teachers’ beliefs and

knowledge on error-handling practices during class discussion of mathematics.

Journal for Research in Mathematics Education, 42, 2-38.

Campbell, P. F., Nishio, M., Smith, T. M., Clark, L. M., Conant, D. L., Rust, A. H.,

. . . Choi, Y. (2014). The relationship between teachers’ mathematical content

and pedagogical knowledge, teachers’ perceptions, and student achievement.

Journal for Research in Mathematics Education, 45, 419-459.

Carpenter, T. P., Fennema, E., Peterson, P. L., & Carey, D. A. (1988). Teachers’

pedagogical content knowledge of students’ problem solving in elementary arith-

metic. Journal for Research in Mathematics Education, 19, 385-401.

Aaronson, D., Barrow, L., & Sander, W. (2007). Teachers and student achievement

in the Chicago public high schools. Journal of Labor Economics, 25, 95-135.

Anderson, R. N., Greene, M. L., & Loewen, P. S. (1988). Relationships among teach-

ers’ and students’ thinking skills, sense of efficacy, and student achievement.

Alberta Journal of Educational Research, 34, 148-165.

Bacher-Hicks, A., Chin, M. J., Hill, H. C., & Staiger, D. O. (2018). Explaining

teacher effects on achievement using commonly found teacher-level predictors.

Manuscript submitted for publication.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching:

What makes it special? Journal of Teacher Education, 59, 389-407.

Bandura, A. (1997). Self-efficacy: The exercise of control. New York, NY: W. H. Freeman.

Baumert, J., Kunter, M., Blum, W., Brunner, M., Voss, T., Jordan, A., . . . Tsai,

Y. M. (2010). Teachers’ mathematical knowledge, cognitive activation in the

classroom, and student progress. American Educational Research Journal, 47,

133-180.

Begle, E. G. (1979). Critical variables in mathematics education: Findings from a

survey of the empirical literature. Washington, DC: Mathematical Association of

America and the Council of Teachers of Mathematics.

Bell, C. A., Wilson, S. M., Higgins, T., & McCoach, D. B. (2010). Measuring the

effects of professional development on teacher knowledge: The case of devel-

oping mathematical ideas. Journal for Research in Mathematics Education, 41,

479-512.

Berman, P., & McLaughlin, M. W. (1977). Federal programs supporting educational

change, Volume VII: Factors affecting implementation and continuation. Santa

Monica, CA: The RAND Corporation.

Boonen, T., Van Damme, J., & Onghena, P. (2014). Teacher effects on student

achievement in first grade: Which aspects matter most? School Effectiveness and

School Improvement, 25, 126-152.

Boyd, D. J., Grossman, P. L., Lankford, H., Loeb, S., & Wyckoff, J. (2009). Teacher

preparation and student achievement. Educational Evaluation and Policy

Analysis, 31, 416-440.

Bray, W. S. (2011). A collective case study of the influence of teachers’ beliefs and

knowledge on error-handling practices during class discussion of mathematics.

Journal for Research in Mathematics Education, 42, 2-38.

Campbell, P. F., Nishio, M., Smith, T. M., Clark, L. M., Conant, D. L., Rust, A. H.,

. . . Choi, Y. (2014). The relationship between teachers’ mathematical content

and pedagogical knowledge, teachers’ perceptions, and student achievement.

Journal for Research in Mathematics Education, 45, 419-459.

Carpenter, T. P., Fennema, E., Peterson, P. L., & Carey, D. A. (1988). Teachers’

pedagogical content knowledge of students’ problem solving in elementary arith-

metic. Journal for Research in Mathematics Education, 19, 385-401.

28.
1130 Educational Policy 33(7)

Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C. P., & Loef, M. (1989).

Using knowledge of children’s mathematics thinking in classroom teaching: An

experimental study. American Educational Research Journal, 26, 499-531.

Charalambous, C. Y., Philippou, G. N., & Kyriakides, L. (2008). Tracing the devel-

opment of preservice teachers’ efficacy beliefs in teaching mathematics during

fieldwork. Educational Studies in Mathematics, 67(2), 125-142.

Charalambous, C. Y., Hill, H. C., McGinn, D., & Chin, M. J. (2017). Teacher knowl-

edge and student learning: Bringing together two different conceptualizations of

teacher knowledge. Manuscript in preparation

Chetty, R., Friedman, J. N., & Rockoff, J. E. (2014). Measuring the impacts of teach-

ers II: Teacher value-added and student outcomes in adulthood. The American

Economic Review, 104, 2633-2679.

Choi, D. (2010). The impact of competing definitions of quality on the geographical

distributions of teachers. Educational Policy, 24, 359-397.

Clotfelter, C. T., Ladd, H. F., & Vigdor, J. L. (2007). Teacher credentials and stu-

dent achievement: Longitudinal analysis with student fixed effects. Economics of

Education Review, 26, 673-682.

Cochran-Smith, M., Cannady, M., McEachern, K., Mitchell, K., Piazza, P., Power,

C., & Ryan, A. (2012). Teachers’ education and outcomes: Mapping the research

terrain. Teachers College Record, 114(10), 1-49.

Darling-Hammond, L. (2010). The Flat World and education: How America’s commit-

ment to equity will determine our future. New York, NY: Teachers College Press.

Dweck, C. S. (2006). Mindset: The new psychology of success. New York, NY:

Random House.

Dweck, C. S., Chiu, C. Y., & Hong, Y. Y. (1995). Implicit theories and their role in

judgments and reactions: A word from two perspectives. Psychological Inquiry,

6, 267-285.

Ernest, P. (1989). The knowledge, beliefs, and attitudes of the mathematics teacher: A

model. Journal of Education for Teaching, 15, 13-33.

Even, R., & Tirosh, D. (2008). Teacher knowledge and understanding of students’

mathematical learning and thinking. In L. D. English (Ed.), Handbook of inter-

national research in mathematics education (2nd ed., pp. 202-223). New York,

NY: Routledge.

Fang, Z. (1996). A review of research on teacher beliefs and practices. Educational

Research, 38, 47-65.

Goldhaber, D., & Hansen, M. (2013). Is it just a bad class? Assessing the long-term

stability of estimated teacher performance. Economica, 80, 589-612.

Goldhaber, D., Lavery, L., & Theobald, R. (2015). Uneven playing field? Assessing

the teacher quality gap between advantaged and disadvantaged students.

Educational Researcher, 44, 293-307. doi:10.3102/0013189X15592622

Goldhaber, D., Quince, V., & Theobald, R. (2016). Reconciling different estimates

of teacher quality gaps based on value added. National Center for Analysis of

Longitudinal Data in Education Research. Retrieved from http://www.calder-

center.org/sites/default/files/Full%20Text_0.pdf

Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C. P., & Loef, M. (1989).

Using knowledge of children’s mathematics thinking in classroom teaching: An

experimental study. American Educational Research Journal, 26, 499-531.

Charalambous, C. Y., Philippou, G. N., & Kyriakides, L. (2008). Tracing the devel-

opment of preservice teachers’ efficacy beliefs in teaching mathematics during

fieldwork. Educational Studies in Mathematics, 67(2), 125-142.

Charalambous, C. Y., Hill, H. C., McGinn, D., & Chin, M. J. (2017). Teacher knowl-

edge and student learning: Bringing together two different conceptualizations of

teacher knowledge. Manuscript in preparation

Chetty, R., Friedman, J. N., & Rockoff, J. E. (2014). Measuring the impacts of teach-

ers II: Teacher value-added and student outcomes in adulthood. The American

Economic Review, 104, 2633-2679.

Choi, D. (2010). The impact of competing definitions of quality on the geographical

distributions of teachers. Educational Policy, 24, 359-397.

Clotfelter, C. T., Ladd, H. F., & Vigdor, J. L. (2007). Teacher credentials and stu-

dent achievement: Longitudinal analysis with student fixed effects. Economics of

Education Review, 26, 673-682.

Cochran-Smith, M., Cannady, M., McEachern, K., Mitchell, K., Piazza, P., Power,

C., & Ryan, A. (2012). Teachers’ education and outcomes: Mapping the research

terrain. Teachers College Record, 114(10), 1-49.

Darling-Hammond, L. (2010). The Flat World and education: How America’s commit-

ment to equity will determine our future. New York, NY: Teachers College Press.

Dweck, C. S. (2006). Mindset: The new psychology of success. New York, NY:

Random House.

Dweck, C. S., Chiu, C. Y., & Hong, Y. Y. (1995). Implicit theories and their role in

judgments and reactions: A word from two perspectives. Psychological Inquiry,

6, 267-285.

Ernest, P. (1989). The knowledge, beliefs, and attitudes of the mathematics teacher: A

model. Journal of Education for Teaching, 15, 13-33.

Even, R., & Tirosh, D. (2008). Teacher knowledge and understanding of students’

mathematical learning and thinking. In L. D. English (Ed.), Handbook of inter-

national research in mathematics education (2nd ed., pp. 202-223). New York,

NY: Routledge.

Fang, Z. (1996). A review of research on teacher beliefs and practices. Educational

Research, 38, 47-65.

Goldhaber, D., & Hansen, M. (2013). Is it just a bad class? Assessing the long-term

stability of estimated teacher performance. Economica, 80, 589-612.

Goldhaber, D., Lavery, L., & Theobald, R. (2015). Uneven playing field? Assessing

the teacher quality gap between advantaged and disadvantaged students.

Educational Researcher, 44, 293-307. doi:10.3102/0013189X15592622

Goldhaber, D., Quince, V., & Theobald, R. (2016). Reconciling different estimates

of teacher quality gaps based on value added. National Center for Analysis of

Longitudinal Data in Education Research. Retrieved from http://www.calder-

center.org/sites/default/files/Full%20Text_0.pdf

29.
Hill et al. 1131

Grubb, W. N. (2008). Multiple resources, multiple outcomes: Testing the “improved”

school finance with NELS88. American Educational Research Journal, 45, 104-

144.

Guarino, C., Dieterle, S. G., Bargagliotti, A. E., & Mason, W. M. (2013). What can

we learn about effective early mathematics teaching? A framework for estimating

causal effects using longitudinal survey data. Journal of Research on Educational

Effectiveness, 6, 164-198.

Hickman, J. J., Fu, J., & Hill, H. C. (2012). Creation and dissemination of upper-

elementary mathematics assessment modules. Princeton, NJ: ETS.

Harris, D. N., & Sass, T. R. (2011). Teacher training, teacher quality and student

achievement. Journal of Public Economics, 95, 798-812.

Helmke, A., & Schrader, F. W. (1987). Interactional effects of instructional qual-

ity and teacher judgement accuracy on achievement. Teaching and Teacher

Education, 3, 91-98.

Hill, H. C., & Chin, M. J. (under review). Connecting teachers’ knowledge of stu-

dents, instruction, and achievement outcomes.

Hill, H. C., Kapitula, L., & Umland, K. (2011). A validity argument approach to

evaluating teacher value-added scores. American Educational Research Journal,

48(3), 794-831.

Hill, H. C., & Lubienski, S. T. (2007). Teachers’ mathematics knowledge for teaching

and school context: A study of California teachers. Educational Policy, 21(5),

747-768.

Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowl-

edge for teaching on student achievement. American Educational Research

Journal, 42, 371-406.

Hoy, W. K., & Woolfolk, A. E. (1993). Teachers’ sense of efficacy and the organiza-

tional health of schools. The Elementary School Journal, 93, 355-372.

Jackson, C. K. (2009). Student demographics, teacher sorting, and teacher quality:

Evidence from the end of school desegregation. Journal of Labor Economics,

27, 213-256.

Justice, L. M., Mashburn, A. J., Hamre, B. K., & Pianta, R. C. (2008). Quality of

language and literacy instruction in preschool classrooms serving at-risk pupils.

Early Childhood Research Quarterly, 23, 51-68.

Kane, T. J., Rockoff, J. E., & Staiger, D. O. (2008). What does certification tell

us about teacher effectiveness? Evidence from New York City. Economics of

Education Review, 27, 615-631.

Kane, T. J., & Staiger, D. O. (2012). Gathering feedback for teaching: Combining high-

quality observations, student surveys, and achievement gains. Seattle, WA: The

Measures of Effective Teaching Project, The Bill and Melinda Gates Foundation.

Koedel, C., & Betts, J. R. (2011). Does student sorting invalidate value-added mod-

els of teacher effectiveness? An extended analysis of the Rothstein critique.

Education Finance and Policy, 6, 18-42.

Lang, L. B., Schoen, R. R., LaVenia, M., & Oberlin, M. (2014). Mathematics

Formative Assessment System—Common Core State Standards: A randomized

Grubb, W. N. (2008). Multiple resources, multiple outcomes: Testing the “improved”

school finance with NELS88. American Educational Research Journal, 45, 104-

144.

Guarino, C., Dieterle, S. G., Bargagliotti, A. E., & Mason, W. M. (2013). What can

we learn about effective early mathematics teaching? A framework for estimating

causal effects using longitudinal survey data. Journal of Research on Educational

Effectiveness, 6, 164-198.

Hickman, J. J., Fu, J., & Hill, H. C. (2012). Creation and dissemination of upper-

elementary mathematics assessment modules. Princeton, NJ: ETS.

Harris, D. N., & Sass, T. R. (2011). Teacher training, teacher quality and student

achievement. Journal of Public Economics, 95, 798-812.

Helmke, A., & Schrader, F. W. (1987). Interactional effects of instructional qual-

ity and teacher judgement accuracy on achievement. Teaching and Teacher

Education, 3, 91-98.

Hill, H. C., & Chin, M. J. (under review). Connecting teachers’ knowledge of stu-

dents, instruction, and achievement outcomes.

Hill, H. C., Kapitula, L., & Umland, K. (2011). A validity argument approach to

evaluating teacher value-added scores. American Educational Research Journal,

48(3), 794-831.

Hill, H. C., & Lubienski, S. T. (2007). Teachers’ mathematics knowledge for teaching

and school context: A study of California teachers. Educational Policy, 21(5),

747-768.

Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowl-

edge for teaching on student achievement. American Educational Research

Journal, 42, 371-406.

Hoy, W. K., & Woolfolk, A. E. (1993). Teachers’ sense of efficacy and the organiza-

tional health of schools. The Elementary School Journal, 93, 355-372.

Jackson, C. K. (2009). Student demographics, teacher sorting, and teacher quality:

Evidence from the end of school desegregation. Journal of Labor Economics,

27, 213-256.

Justice, L. M., Mashburn, A. J., Hamre, B. K., & Pianta, R. C. (2008). Quality of

language and literacy instruction in preschool classrooms serving at-risk pupils.

Early Childhood Research Quarterly, 23, 51-68.

Kane, T. J., Rockoff, J. E., & Staiger, D. O. (2008). What does certification tell

us about teacher effectiveness? Evidence from New York City. Economics of

Education Review, 27, 615-631.

Kane, T. J., & Staiger, D. O. (2012). Gathering feedback for teaching: Combining high-

quality observations, student surveys, and achievement gains. Seattle, WA: The

Measures of Effective Teaching Project, The Bill and Melinda Gates Foundation.

Koedel, C., & Betts, J. R. (2011). Does student sorting invalidate value-added mod-

els of teacher effectiveness? An extended analysis of the Rothstein critique.

Education Finance and Policy, 6, 18-42.

Lang, L. B., Schoen, R. R., LaVenia, M., & Oberlin, M. (2014). Mathematics

Formative Assessment System—Common Core State Standards: A randomized

30.
1132 Educational Policy 33(7)

field trial in kindergarten and first grade. Paper presented at the Annual Meeting

of the Society for Research on Educational Effectiveness, Washington, DC.

Lankford, H., Loeb, S., & Wyckoff, J. (2002). Teacher sorting and the plight of urban

schools: A descriptive analysis. Educational Evaluation and Policy Analysis, 24,

37-62.

Lavy, V. (2009). Performance pay and teachers’ effort, productivity, and grading eth-

ics. The American Economic Review, 99, 1979-2011.

Liu, E., & Johnson, S. M. (2006). New teachers’ experiences of hiring: Late, rushed,

and information-poor. Educational Administration Quarterly, 42, 324-360.

Lockwood, J. R., McCaffrey, D. F., Hamilton, L. S., Stecher, B., Le, V. N., & Martinez,

J. F. (2007). The sensitivity of value-added teacher effect estimates to different

mathematics achievement measures. Journal of Educational Measurement, 44,

47-67.

Loeb, S., & Reininger, M. (2004). Public policy and teacher labor markets: What

we know and why it matters. East Lansing, MI: The Education Policy Center at

Michigan State University.

Lynch, K., Chin, M., & Blazar, D. (2017). Relationships between observations of ele-

mentary mathematics instruction and student achievement: Exploring variability

across districts. American Journal of Education, 123(4), 615-646.

Max, J., & Glazerman, S. (2014). Do disadvantaged students get less effective teach-

ing? Key findings from recent Institute of Education Sciences studies. National

Center for Education Evaluation and Regional Assistance. Retrieved from https://

ies.ed.gov/ncee/pubs/20144010/pdf/20144010.pdf

McIlveen, P., & Perera, H. N. (2016). Career optimism mediates the effect of person-

ality on teachers’ career engagement. Journal of Career Assessment, 24, 623-636.

Merton, R. K. (1968). The Matthew effect in science. Science, 159, 56-63.

Metzler, J., & Woessmann, L. (2012). The impact of teacher subject knowledge on

student achievement: Evidence from within-teacher within-student variation.

Journal of Development Economics, 99, 486-496.

Molden, D. C., & Dweck, C. (2006). Meaning in psychology: A lay theories

approach to self-regulation, social perception, and social development. American

Psychologist, 61, 192-203.

Monk, D. H. (1994). Subject area preparation of secondary mathematics and science

teachers and student achievement. Economics of Education Review, 13, 125-145.

Mullens, J. E., Murnane, R. J., & Willett, J. B. (1996). The contribution of training

and subject matter knowledge to teaching effectiveness: A multilevel analysis of

longitudinal evidence from Belize. Comparative Education Review, 40, 139-157.

Palardy, G. J., & Rumberger, R. W. (2008). Teacher effectiveness in first grade: The

importance of background qualifications, attitudes, and instructional practices for

student learning. Educational Evaluation and Policy Analysis, 30, 111-140.

Papay, J. (2011). Different tests, different answers: The stability of teacher value-

added estimates across outcomes measures. American Educational Research

Journal, 48, 163-193.

field trial in kindergarten and first grade. Paper presented at the Annual Meeting

of the Society for Research on Educational Effectiveness, Washington, DC.

Lankford, H., Loeb, S., & Wyckoff, J. (2002). Teacher sorting and the plight of urban

schools: A descriptive analysis. Educational Evaluation and Policy Analysis, 24,

37-62.

Lavy, V. (2009). Performance pay and teachers’ effort, productivity, and grading eth-

ics. The American Economic Review, 99, 1979-2011.

Liu, E., & Johnson, S. M. (2006). New teachers’ experiences of hiring: Late, rushed,

and information-poor. Educational Administration Quarterly, 42, 324-360.

Lockwood, J. R., McCaffrey, D. F., Hamilton, L. S., Stecher, B., Le, V. N., & Martinez,

J. F. (2007). The sensitivity of value-added teacher effect estimates to different

mathematics achievement measures. Journal of Educational Measurement, 44,

47-67.

Loeb, S., & Reininger, M. (2004). Public policy and teacher labor markets: What

we know and why it matters. East Lansing, MI: The Education Policy Center at

Michigan State University.

Lynch, K., Chin, M., & Blazar, D. (2017). Relationships between observations of ele-

mentary mathematics instruction and student achievement: Exploring variability

across districts. American Journal of Education, 123(4), 615-646.

Max, J., & Glazerman, S. (2014). Do disadvantaged students get less effective teach-

ing? Key findings from recent Institute of Education Sciences studies. National

Center for Education Evaluation and Regional Assistance. Retrieved from https://

ies.ed.gov/ncee/pubs/20144010/pdf/20144010.pdf

McIlveen, P., & Perera, H. N. (2016). Career optimism mediates the effect of person-

ality on teachers’ career engagement. Journal of Career Assessment, 24, 623-636.

Merton, R. K. (1968). The Matthew effect in science. Science, 159, 56-63.

Metzler, J., & Woessmann, L. (2012). The impact of teacher subject knowledge on

student achievement: Evidence from within-teacher within-student variation.

Journal of Development Economics, 99, 486-496.

Molden, D. C., & Dweck, C. (2006). Meaning in psychology: A lay theories

approach to self-regulation, social perception, and social development. American

Psychologist, 61, 192-203.

Monk, D. H. (1994). Subject area preparation of secondary mathematics and science

teachers and student achievement. Economics of Education Review, 13, 125-145.

Mullens, J. E., Murnane, R. J., & Willett, J. B. (1996). The contribution of training

and subject matter knowledge to teaching effectiveness: A multilevel analysis of

longitudinal evidence from Belize. Comparative Education Review, 40, 139-157.

Palardy, G. J., & Rumberger, R. W. (2008). Teacher effectiveness in first grade: The

importance of background qualifications, attitudes, and instructional practices for

student learning. Educational Evaluation and Policy Analysis, 30, 111-140.

Papay, J. (2011). Different tests, different answers: The stability of teacher value-

added estimates across outcomes measures. American Educational Research

Journal, 48, 163-193.

31.
Hill et al. 1133

Papay, J. P., & Kraft, M. A. (2015). Productivity returns to experience in the teacher

labor market: Methodological challenges and new evidence on long-term career

improvement. Journal of Public Economics, 130, 105-119.

Rice, J. K. (2003). Teacher quality: Understanding the effectiveness of teacher attri-

butes. Washington, DC: Economic Policy Institute.

Rockoff, J. E., Jacob, B. A., Kane, T. J., & Staiger, D. O. (2011). Can you recognize

an effective teacher when you recruit one? Education Finance and Policy, 6,

43-74.

Rose, J. S., & Medway, F. J. (1981). Measurement of teachers’ beliefs in their control

over student outcome. The Journal of Educational Research, 74, 185-190.

Rotter, J. B. (1966). Generalized expectancies for internal versus external control of

reinforcement. Psychological Monographs: General and Applied, 80, 1-28.

Rowan, B., Correnti, R., & Miller, R. (2002). What large-scale survey research tells

us about teacher effects on student achievement: Insights from the prospects

study of elementary schools. The Teachers College Record, 104, 1525-1567.

Roza, M., & Miller, R. (2009). Separation of degrees: State-by-state analysis of

teacher compensation for Master’s degrees. Seattle, WA: Center on Reinventing

Public Education.

Rutledge, S. A., Harris, D. N., Thompson, C. T., & Ingle, W. K. (2008). Certify, blink,

hire: An examination of the process and tools of teacher screening and selection.

Leadership and Policy in Schools, 7, 237-263.

Schleicher, A. & Organisation for Economic Co-Operation and Development.

(2014). Equity, excellence and inclusiveness in education: Policy lessons from

around the world. Paris, France: Organisation for Economic Co-Operation and

Development.

Schultz, L. M. (2014). Inequitable dispersion: Mapping the distribution of highly

qualified teachers in St. Louis metropolitan elementary schools. Education

Policy Analysis Archives, 22(90), 1-20.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching.

Educational Researcher, 15(2), 4-14.

Sleeter, C. (2014). Toward teacher education research that informs policy. Educational

Researcher, 43, 146-153.

Soodak, L. C., & Podell, D. M. (1996). Teacher efficacy: Toward the understanding

of a multi-faceted construct. Teaching and Teacher Education, 12, 401-411.

Stipek, D. (2012). Context matters: Effects of student characteristics and perceived

administrative and parental support on teacher self-efficacy. The Elementary

School Journal, 112, 590-606.

Stronge, J. H., Ward, T. J., & Grant, L. W. (2011). What makes good teachers good?

A cross-case analysis of the connection between teacher effectiveness and stu-

dent achievement. Journal of Teacher Education, 62, 339-355.

Tatto, M. T., Schwille, J., Senk, S., Ingvarson, L., Peck, R., & Rowley, G. (2008).

Teacher Education and Development Study in Mathematics (TEDS-M): Policy,

practice, and readiness to teach primary and secondary mathematics. Conceptual

Papay, J. P., & Kraft, M. A. (2015). Productivity returns to experience in the teacher

labor market: Methodological challenges and new evidence on long-term career

improvement. Journal of Public Economics, 130, 105-119.

Rice, J. K. (2003). Teacher quality: Understanding the effectiveness of teacher attri-

butes. Washington, DC: Economic Policy Institute.

Rockoff, J. E., Jacob, B. A., Kane, T. J., & Staiger, D. O. (2011). Can you recognize

an effective teacher when you recruit one? Education Finance and Policy, 6,

43-74.

Rose, J. S., & Medway, F. J. (1981). Measurement of teachers’ beliefs in their control

over student outcome. The Journal of Educational Research, 74, 185-190.

Rotter, J. B. (1966). Generalized expectancies for internal versus external control of

reinforcement. Psychological Monographs: General and Applied, 80, 1-28.

Rowan, B., Correnti, R., & Miller, R. (2002). What large-scale survey research tells

us about teacher effects on student achievement: Insights from the prospects

study of elementary schools. The Teachers College Record, 104, 1525-1567.

Roza, M., & Miller, R. (2009). Separation of degrees: State-by-state analysis of

teacher compensation for Master’s degrees. Seattle, WA: Center on Reinventing

Public Education.

Rutledge, S. A., Harris, D. N., Thompson, C. T., & Ingle, W. K. (2008). Certify, blink,

hire: An examination of the process and tools of teacher screening and selection.

Leadership and Policy in Schools, 7, 237-263.

Schleicher, A. & Organisation for Economic Co-Operation and Development.

(2014). Equity, excellence and inclusiveness in education: Policy lessons from

around the world. Paris, France: Organisation for Economic Co-Operation and

Development.

Schultz, L. M. (2014). Inequitable dispersion: Mapping the distribution of highly

qualified teachers in St. Louis metropolitan elementary schools. Education

Policy Analysis Archives, 22(90), 1-20.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching.

Educational Researcher, 15(2), 4-14.

Sleeter, C. (2014). Toward teacher education research that informs policy. Educational

Researcher, 43, 146-153.

Soodak, L. C., & Podell, D. M. (1996). Teacher efficacy: Toward the understanding

of a multi-faceted construct. Teaching and Teacher Education, 12, 401-411.

Stipek, D. (2012). Context matters: Effects of student characteristics and perceived

administrative and parental support on teacher self-efficacy. The Elementary

School Journal, 112, 590-606.

Stronge, J. H., Ward, T. J., & Grant, L. W. (2011). What makes good teachers good?

A cross-case analysis of the connection between teacher effectiveness and stu-

dent achievement. Journal of Teacher Education, 62, 339-355.

Tatto, M. T., Schwille, J., Senk, S., Ingvarson, L., Peck, R., & Rowley, G. (2008).

Teacher Education and Development Study in Mathematics (TEDS-M): Policy,

practice, and readiness to teach primary and secondary mathematics. Conceptual

32.
1134 Educational Policy 33(7)

Framework. East Lansing: Teacher Education and Development International

Study Center, College of Education, Michigan State University.

Tschannen-Moran, M., & Hoy, A. W. (2001). Teacher efficacy: Capturing an elusive

construct. Teaching and Teacher Education, 17, 783-805.

Tschannen-Moran, M., Hoy, A. W., & Hoy, W. K. (1998). Teacher efficacy: Its

meaning and measure. Review of Educational Research, 68, 202-248.

Wayne, A. J., & Youngs, P. (2003). Teacher characteristics and student achievement

gains: A review. Review of Educational Research, 73, 89-122.

Author Biographies

Heather C. Hill is the Jerome T. Murphy Professor in Education at the Harvard

Graduate School of Education. Her primary work focuses on teacher and teaching

quality and the effects of policies aimed at improving both. She is also known for

developing instruments for measuring teachers’ mathematical knowledge for teaching

(MKT) and the mathematical quality of instruction (MQI) within classrooms.

Charalambos Y. Charalambous is an Assistant Professor in Educational Research

and Evaluation at the Department of Education of the University of Cyprus. His main

research interests center on issues of teaching/teacher effectiveness, contributors to

instructional quality, and the effects of instruction on student learning.

Mark J. Chin is a PhD Candidate in Education Policy and Program Evaluation at

Harvard University. His research interests center on the experiences of students of

color and first-and second-generation students in US K-12 educational contexts. His

work focuses on explorations of how race, racism, and assimilative pressures cause

inequality in these students’ outcomes.

Framework. East Lansing: Teacher Education and Development International

Study Center, College of Education, Michigan State University.

Tschannen-Moran, M., & Hoy, A. W. (2001). Teacher efficacy: Capturing an elusive

construct. Teaching and Teacher Education, 17, 783-805.

Tschannen-Moran, M., Hoy, A. W., & Hoy, W. K. (1998). Teacher efficacy: Its

meaning and measure. Review of Educational Research, 68, 202-248.

Wayne, A. J., & Youngs, P. (2003). Teacher characteristics and student achievement

gains: A review. Review of Educational Research, 73, 89-122.

Author Biographies

Heather C. Hill is the Jerome T. Murphy Professor in Education at the Harvard

Graduate School of Education. Her primary work focuses on teacher and teaching

quality and the effects of policies aimed at improving both. She is also known for

developing instruments for measuring teachers’ mathematical knowledge for teaching

(MKT) and the mathematical quality of instruction (MQI) within classrooms.

Charalambos Y. Charalambous is an Assistant Professor in Educational Research

and Evaluation at the Department of Education of the University of Cyprus. His main

research interests center on issues of teaching/teacher effectiveness, contributors to

instructional quality, and the effects of instruction on student learning.

Mark J. Chin is a PhD Candidate in Education Policy and Program Evaluation at

Harvard University. His research interests center on the experiences of students of

color and first-and second-generation students in US K-12 educational contexts. His

work focuses on explorations of how race, racism, and assimilative pressures cause

inequality in these students’ outcomes.