Teacher Characteristics and Student Learning in Mathematics

Contributed by:
Jonathan James
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.
1. EPXXXX10.1177/0895904818755468Educational PolicyHill et al.
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
in Mathematics: journals.sagepub.com/home/epx
A Comprehensive
Heather C. Hill1, Charalambos Y. Charalambous2,
and Mark J. Chin1
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
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
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
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
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
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
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.
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
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
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
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:
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
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:
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
•• 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 ;
•• 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
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.
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.
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
†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,
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
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
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
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,
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
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
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.
27. Hill et al. 1129
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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.