Pulling Back the Curtain: Revealing the Cumulative Importance of High-Performing,Highly Qualified Teachers on Students’ Educational Outcome

Abstract

This study examines the relationship between two dominant measures of teacher quality, teacher qualification and teacher effectiveness (measured by value-added modeling), in terms of their influence on students’ short-term academic growth and long-term educational success (measured by bachelor’s degree attainment). As students are exposed to teachers of varying quality over the course of their schooling, this study computes cumulative teacher quality indices that are able to more precisely estimate the impact of teacher quality. Notably, this study found that students who had been taught by a succession of high-performing and qualified teachers tend to have a positive relationship with students’ short- and long-term educational success.

Keywords

teacher qualification teacher effectiveness value-added modeling student achievement cumulative teacher quality

Introduction

Beginning with the influential Coleman Report released in 1966, a growing body of research has repeatedly confirmed that teacher quality, compared with other commonly measured school-level factors such as class size, curriculum, school climate, and technology, is the most influential factor in students’ educational outcomes (Coleman et al., 1966; Hanushek & Rivkin, 2006). With this body of research in mind, it seemed self-evident by the turn of the century that schools would need to be staffed by high-quality teachers if they were to meet the needs of diverse students while promoting challenging and engaging instruction (Borman & Kimball, 2005).

However, although teachers undoubtedly play a significant role in the educational success of their students (Goldhaber, 2007; Rivkin, Hanushek, & Kain, 2005), not all students are exposed to high-quality teachers. Regardless of how one defines teacher quality, historically disadvantaged students are less likely to be taught by high-quality teachers—as measured by level of education, years of experience, or value-added measures (Kalogrides & Loeb, 2013). Even more disturbingly, disadvantaged students are more likely to be taught by low-quality teachers every year (Goldhaber, Quince, & Theobald, 2018). This inequity—being taught by low-quality teachers—can be accumulative as disadvantaged students progress through their grades, which, in turn, can have a serious impact on a student’s educational experience and success.

Therefore, considering the cumulative and accumulative nature of education, in a system in which students are exposed to a dozen or more teachers during their schooling, more attention is needed to understand the cumulative importance of high-performing, highly qualified teachers in students’ educational outcomes (DiPrete & Eirich, 2006). We know much less about the consequences of being taught by low-performing, less qualified teachers sequentially, in part, because we tend to focus on a single teacher’s contribution while neglecting the cumulative nature of education (Potter & Morris, 2017). This cumulative approach aligns with a study by Sanders and Rivers (1996), who first introduced the idea of examining the cumulative effect of a student’s exposure to high or low value-added teachers over three consecutive years. Therefore, to better understand the significant contribution that teachers have on our students’ educational pathways and how this accumulation of experience results in educational success, this study examines the relationship between teacher quality—qualifications and effectiveness—and students’ educational success in a cumulative manner. Although there are various measures of teacher quality, this study focuses on two major indicators—teacher qualifications and teacher effectiveness—largely because of their prevalence in existing policy and research. In this study, “effectiveness” reflects a teacher’s contribution to growth in student achievement, whereas “qualifications” refers to a teacher’s years of experience, level of education, and subject-matter expertise.

Furthermore, although the previous literature has demonstrated the many ways in which teachers contribute to students’ intellectual and emotional growth, such research has focused on short-term outcomes, typically student test scores. Notably, teachers’ effectiveness has been shown to affect not only students’ short-term academic achievement but also other desirable outcomes: studies have shown, for instance, that such students are more likely to attend college and earn a higher salary (Chetty, Friedman, & Rockoff, 2013a, 2013b) if taught by effective teachers. Given that the enduring impact of teacher quality may emerge slowly or may be observable only when examined over time (Chetty et al., 2013b; Sanders & Rivers, 1996), this study also examines the contribution of being sequentially taught by high-performing, qualified teachers on students’ educational success in the long term, particularly in terms of bachelor’s degree attainment.

By using the Longitudinal Study of American Youth (LSAY) data set and taking the above approaches and perspectives into account, the overarching goal of this study is to explore the following research questions:

Research Question 1: What is the relationship between cumulative teachers’ qualifications—as measured by degrees, subject-matter expertise, and years of experience—and cumulative teachers’ effectiveness on students’ short-term academic growth?

Research Question 2: What is the relationship between cumulative teachers’ qualifications—as measured by degrees, subject-matter expertise, and years of experience—and cumulative teachers’ effectiveness on students’ long-term education attainment?

Literature Review

Research on Teachers’ Value-Added Measure

Value-added measures are used in education to provide estimates of teacher effectiveness on students’ academic growth and, consequently, to provide policymakers with a measure of how much “value” an individual teacher “adds” to student achievement over time. With their use of longitudinal and extensive state-level data sets, numerous studies have demonstrated methodological improvements to value-added modeling (VAM) and the approach’s contribution to the field of education (Goldhaber, 2007; Jacob, Lefgren, & Sims, 2008; Rivkin et al., 2005).

Research has shown that if students encounter highly effective teachers year after year, then the cumulative and enduring effect on student achievement will likely be significantly higher than for students who miss out on such experiences (Sanders & Rivers, 1996). Sanders and Rivers (1996) concluded that even students with comparable initial performance levels can have “vastly different academic outcomes as a result of the sequence of teachers to which they are assigned” (p. 9). Research in Dallas found similar results by comparing students taught sequentially by highly effective teachers with students taught sequentially by ineffective teachers. The two groups of students had a significantly large and disturbing achievement gap mainly because of the teachers they had experienced (Archer, 1998; Haycock, 1998).

In addition, research has found that the effect of high value-added teachers on improving student achievement is three times that of reducing class size (Nye, Konstantopoulos, & Hedges, 2004). Moreover, the literature shows that teachers with higher value-added scores tend to have a positive influence on students’ noncognitive outcomes, such as attendance, behavior, and graduation (Jackson, 2012). Although a teacher’s effect may fade over time (Kane & Staiger, 2008), studies have found not only that a teacher’s effect can be detected a few years later (Sanders & Rivers, 1996) but also that they have an enduring effect on students’ long-term outcomes (Chetty et al., 2010; Chetty, Friedman, & Rockoff, 2011). In particular, recent research has documented that a high value-added teacher not only has a positive effect on a student’s short-term achievement but also has a positive effect on long-term outcomes, such as earnings, employment, and college attainment (Chetty et al., 2013a, 2013b). Proponents of VAM also note that classroom observations do not necessarily capture student learning but, rather, portray what teachers are teaching in class (Dynarski, 2016). Although classroom observation provides meaningful insights into teaching, failing to recognize, acknowledge, or differentiate effective teaching (i.e., great teaching to good) has been criticized (Weisberg, Sexton, Mulhern, & Keeling, 2009). Thus, not surprisingly, the idea of measuring how much value a teacher adds to student achievement has drawn a great deal of attention and consideration in recent decades.

Despite its conceptual and methodological appeal, VAM has its share of skeptics who raise questions about the validity and reliability of the methods. For example, VAM takes into account the challenges that are associated with student–teacher sorting bias by accounting for prior test scores in fixed-effects models (Guarino, Maxfield, Reckase, Thompson, & Wooldridge, 2015; Harris & Sass, 2011). However, critics argue that VAM may not efficiently and effectively mitigate student–teacher selection bias (Rothstein, 2009, 2017), producing lower ratings for teachers with more disadvantaged students and with relatively low-performing students (Collins & Amrein-Beardsley, 2014; Kane, 2017). Simultaneously, the VAM scores for a teacher fluctuate markedly from year to year, undermining the trustworthiness of the model’s results and lowering confidence in its use in the implementation of high-stakes policies concerning teachers (Harris, 2011). RAND researchers assert that based on the standardized tests used, different conclusions regarding the effectiveness of a teacher may be drawn (Koretz, 2008; McCaffrey, Lockwood, Koretz, & Hamilton, 2003). Papay’s (2011) study examined this concern and noted that

policymakers and practitioners who wish to use these estimates to make high-stakes decisions must think carefully about the consequences of these differences, recognizing that even decisions seemingly as arbitrary as when to schedule the test within the school year will likely produce variation in teacher effectiveness estimates. (p. 188)

Overall, these putative benefits and challenges regarding VAM pose a dilemma for many educational leaders who must decide whether to use the method in their school and education system—not an easy choice, as noted by the Economic Policy Institute: “While there are good reasons for concern about the current system of teacher evaluation, there are also good reasons to be concerned about claims that measuring teachers’ effectiveness largely by student test scores will lead to improved student achievement” (Baker et al., 2010, p. 1).

Research on Teacher Qualification

For some time, teacher qualification measures have significantly influenced key teacher policies such as hiring, tenure, the salary schedule, and compensation. Although scholars debate the utility of these teacher qualifications, many school districts base their salary schedules and compensation policies—which constitute a substantial portion of overall education budgets—on a teacher’s years of experience and level of education.

Years of Experience

The number of years of experience is, by far, the most important proxy of teacher qualification, playing a key role in determining the salary schedule and tenure for many teachers. Theoretically, in most fields, gaining years of experience means having better skills, expertise in and knowledge of one’s profession; for this reason, many occupations, including education, have favored a person with more experience. However, counterintuitively, the literature has found mixed results regarding the effectiveness of teachers’ years of experience on student achievement.

On one hand, research has found that teacher experience has a positive relation with students’ academic achievement (Clotfelter, Ladd, & Vigdor, 2007; Ladd, 2008). Even comparing teachers with advanced degrees, teaching experience has a significant correlation with student achievement (Grissmer, Flanagan, Kawata, Williamson, & LaTourrette, 2000), not only in the immediate classroom but also on the long-term educational outcomes of students. In one study, Chetty and colleagues (2010) found that kindergarten students who were taught by more experienced teachers had higher achievement and earnings as adults. Recently, Papay and Kraft (2015) found that teachers seem to improve their productivity continually throughout their careers.

On the other hand, despite these promising results, most of the literature has shown that the number of years that a teacher has taught translates to minimal or limited gains in student achievement (Aaronson, Barrow, & Sander, 2007; Goe, 2007). Intriguingly, most gains in effectiveness occur during the first few years of the teacher’s career, but seem to flatten afterward (Chingos & Peterson, 2011; Rockoff, 2004). In other words, as far as a teacher’s level of experience matters, most studies suggest that teachers improve significantly in their early careers but minimally thereafter.

Advanced Degree and Out-of-Field Teaching

As more teachers obtain graduate degrees for a variety of purposes including monetary incentives, there is a growing interest in whether these degrees are positively associated with student achievement. Over the past 50 years, the number of teachers with a master’s degree has almost doubled, and across the United States, schools and states are spending more than US$14 billion on compensating teachers who hold advanced degrees (Miller & Roza, 2012). However, relatively little is known about the effects and benefits that advanced degrees offer. Although few studies have confirmed the positive effect of teachers’ advanced degrees on student achievement (Betts, Zau, & Rice, 2003; Goldhaber & Brewer, 1997), many scholars have found that a teacher’s advanced degree attainment has little or insignificant effect on student performance (Hanushek, Kain, O’Brien, & Rivkin, 2005; Ladd & Sorensen, 2015).

This inconsistency is further complicated by the fact that many studies also neglect to consider whether teachers hold an advanced degree in the field in which they are teaching. A significant number of studies consider only the degree level (bachelor’s, master’s, or higher) without considering whether that degree is in their subject specialty (Goe, 2007). Given that, each year, over one fifth of public 7th- to 12th-grade teachers teach subjects that are out-of-field (Ingersoll, 2003), simply examining the degree level is not sufficient to understand the true effect of a teacher’s advanced degree. Several studies indicate that teachers who have an advanced degree in the subject they teach tend to have an association with higher student achievement in middle and high school (Goldhaber & Brewer, 1996, 1997, 2000). Although research in this area is scarce, studies have shown that students who were taught by teachers who majored in their subject-area regularly performed better than students who were taught by teachers who majored in something else (Dee & Cohodes, 2008).

The Present Study

Although research shows that teacher quality matters in students’ educational success, the literature is still limited to identifying which proxies or sets of teacher quality attributes best contribute to desired educational outcomes (Hanushek & Rivkin, 2006; Rice, 2003). Fortunately, in recent decades, research has moved beyond finding a single measure of effective teacher quality (i.e., teachers’ value-added estimates, teacher qualifications) or criticizing one another’s logistics and methodology toward understanding how a combination of measures of teacher quality can be used to identify and support our teachers.

Among several important studies, the measures of effective teaching (MET) project provides substantive guidance regarding fruitful ways to think about, use, and combine these measures of teacher quality. The MET study reports that a combination of measures such as classroom observation, teachers’ value-added, and student surveys can in fact help identify and foster improved teaching. Instead of using each measure in isolation, a combination of them is more likely to identify teachers who perform better and teachers who need targeted professional development (Kane & Cantrell, 2013). Moreover, emerging research has begun to reveal how these measures are positively related to one another. For example, Cowan and Goldhaber (2016) recently found positive and significant relationships between teachers certified by the National Board for Professional Teaching Standard and teachers’ value-added measures.

Unfortunately, none of the research has examined and compared various proxies of teacher quality in a single study, in a cumulative manner, and in relation to both the short- and long-term outcomes of students. Research in sociology and public policy has examined and measured social inequality from a cumulative perspective, but less attention has been paid to the educational setting (Potter & Roksa, 2013; Willson, Shuey, & Elder, 2007). Thus, to better understand the relationship between teacher quality and students’ educational success, this study proposes a thorough examination of the significance of cumulative teachers’ quality—as measured by qualification and effectiveness—on both the short- and long-term outcomes of students. By examining the efficacy of the teacher quality measures currently and widely used in education, this study intends to contribute to the long-standing debate regarding inequitable access to teacher quality.

Method

Data

To examine the relationship between cumulative teachers’ quality and students’ educational success, this study used the LSAY data set, which is a national probability sample funded by the National Science Foundation (Miller, Kimmel, Hoffer, & Nelson, 2000). Between 1987 and 1994, the LSAY initially collected survey data from approximately 6,000 students in public high schools throughout the United States as well as from their parents, teachers, and school administrators. The sample was divided into two groups, a younger cohort (7th grade) and an older cohort (10th grade), in which most of the younger cohort attended middle schools that served as feeder schools for the same high schools as the older cohort.

The LSAY selected 51 representative high schools for its data collection based on geographical region (northeast, midwest, south, and west) and community type (urban, suburban, and rural). Using a stratified probability sample, approximately 60 students were selected from each school and for each cohort, and individuals were asked to participate. From this initial selection stage, 5,945 students from 51 secondary schools opted to participate in the study, consisting of 3,116 students in the younger cohort and 2,829 students in the older cohort. To investigate the students’ educational and occupational outcomes, data collection then resumed in 2007, when the participants were 33 to 37 years of age.

The students in the LSAY not only completed survey questionnaires each year but also took standardized mathematics achievement tests during the fall of each year. Each mathematics assessment consisted of items from the National Assessment of Educational Progress (NAEP) and was designed to measure basic skills, algebra, geometry, and quantitative literacy.

In addition to having a rich sample of students, the LSAY also surveyed the mathematics teachers in the participating schools, including teachers who were not teaching the participating students. Among the 1,330 teachers in the participating middle and high schools, a total of 1,018 teachers responded to the survey questionnaires, including 564 mathematics teachers (194 in the middle schools and 370 in the high schools). This sample included not only established teachers but also new incoming teachers.

Moreover, as students progress to upper-level grades, they are less likely to take mathematics classes, compared with students in lower grades in high school. For example, 12th-grade students are less likely to take mathematics courses compared with 9th-grade students. This eventually leads to lower student–teacher ratios as the students progressed (see Table 1). It is important to recognize this aspect because the value-added methods employed in this study calculate teacher effectiveness based on the information of the students in the classroom. Therefore, a small student-to-teacher ratio can cause noise in the estimation of teacher effectiveness. To account for this noise, this study employs the Bayesian shrinkage method (discussed in detail in the “Cumulative Teacher Effectiveness Index” section) (Carlin & Louis, 2000; Papay, 2011).

Table 1

Student–Teacher Ratio in Each Grade (Mathematics)

Grade	Younger cohort		Average student per teacher	Older cohort			Combined
Grade	Number of teachers	Number of students	Average student per teacher	Number of teachers	Number of students	Number of teachers		Number of students
7	196	3,093	15.78				196	3,093
8	205	2,523	12.30				205	2,523
9	325	2,083	6.40				325	2,083
10	376	1,878	4.99	414	2,673	6.45	566	4,551
11	364	1,552	4.26	339	1,639	4.83	517	3,191
12	279	969	3.47	279	1,047	3.75	416	2,016

Note. As younger cohort students feed into older cohorts, there were several teachers who have taught both. Thus, the number of teachers in the “Combined” cell represents an unduplicated number of teachers.

Compared with other national representative data sets, such as the National Education Longitudinal Study of 1988, the Education Longitudinal Study of 2002, or the High School Longitudinal Study of 2009, the LSAY is distinctive in the scope of its data. To the best of my knowledge, it is the only national data set to include annual measures of students’ mathematics achievement alongside teacher information, allowing for a calculation of the cumulative impact of teacher effectiveness and qualifications. Moreover, the two stages of data, which cover almost three decades, allow for a more accurate and comprehensive evaluation of the impact of teacher quality on students’ long-term outcomes.

Because of this uniqueness, previous studies have used the LSAY to examine the factors influencing students’ educational pathways and vocational choice from high school to college and even throughout their careers. These factors include parental influence (Hong, 2010; Wang & Wildman, 1996), teachers’ contribution (Lee, Min, & Mamerow, 2015; Ma, 2001), students’ mathematics attitude and self-efficacy (George, 2003; Ma & Xu, 2004), and other school and extracurricular factors (Betts & Shkolnik, 2000; Gamoran, 2002).

Analytic Sample

The present study used students’ information from both cohorts for analysis (consisting of 3,116 students in the younger cohort and 2,829 students in the older cohort). The paper first restricted the sample to students who have participated in both the 12th-grade survey and the 2007 follow-up survey. This limited the analytic sample to 5,052, which consisted of 2,561 students in the younger cohort and 2,491 students in the older cohort.

In addition, because the key predictor in this study is the teachers’ quality, this study focused on students who have complete information on their mathematics teachers during the data collection. Students may not have this information because they opted out, moved to different high schools, or their teachers refused to participate in the survey. These factors, accounting for approximately 20% of the observations, yielded 3,876 students for analysis (1,807 from the younger cohort and 2,069 from the older cohort).

No more than 5% of cases had little or missing data. To determine whether the missing observations were completely random, this study used Little’s (1988) missing completely at random (MCAR) test. The findings suggested (χ² = 1,443.94, p < .001) the use of missing imputation (MI) to appropriately account for “missingness” in the data set (Little & Rubin, 1989; Schafer & Graham, 2002). Thus, the present study employed MI by chained equations to maximize the use of available information and to minimize bias (Royston, 2005). MI is widely regarded as the most rigorous method for addressing missing data in social science settings (Graham, Hofer, Donaldson, MacKinnon, & Schafer, 1997). In addition to addressing these missing values, the MI method is well suited to this data set because it can compensate for any bias and power that may have been caused by missing observations (Schafer & Graham, 2002).

The model for multiple imputation was performed on both the independent and dependent variables (Royston, 2005; von Hippel, 2007). Five imputations were generated in this study based on the relative efficiency calculation by Rubin (1987). It is important to note that the study not only presents results using MI on both independent and dependent variables but also reanalyzes the data using MI only on independent variables and the complete cases (list-wise deletion) approach. Regardless of which approach is used, the findings were not different and showed the same trends and the same narratives.

Measures

In this study, there are two dependent variables—12th-grade achievement and college degree completion—and two main independent variables—a Cumulative Teacher Effectiveness Index and a Cumulative Teacher Qualification Index. To aid in the interpretation of the findings, the primary independent variables and student achievement in 12th grade were standardized to have a mean 0 and standard deviation (SD) of 1. This makes it possible to interpret the findings as an effect size and enables the study to make comparisons across models and outcome measures (e.g., Hoxby, 2000; McEwan, 2003).

Dependent Variables

Short-Term Achievement

The relationship between cumulative teachers’ quality and students’ short-term achievement is examined using students’ 12th-grade mathematics achievement. Academic achievement in mathematics was measured using tests constructed from the item pools of the NAEP. To establish comparable scores in Cohort 1 and Cohort 2, scores were recalibrated using multiple group item-response theory (Miller et al., 2000); thus, the scores have a range from 1 to 100, with a mean of 50 and a SD of 10.

Long-Term Educational Attainment

The relationship between cumulative teachers’ quality and students’ long-term educational attainment is measured by assaying whether students earned a bachelor’s degree as a dichotomous dependent variable.

Primary Independent Variables

Cumulative Teacher Effectiveness Index

To calculate the teacher’s value-added score, the study uses students’ annual standardized test scores (in mathematics) collected over a six-year period, using post- and pretest scores to isolate the effect of teacher contributions on academic achievement.

Using annual data capturing students’ achievement, the model calculates teacher’s value-added scores in the following manner:

\begin{array}{l} Y_{i j s t} = δ_{0} + Y_{i j s_{(t - 1)}} δ_{1} + {(Y_{j s (t - 1)})}^{2} δ_{2} \\ + {(Y_{j s (t - 1)})}^{3} δ_{3} + χ_{i j s} δ_{4} + γ_{j s} + ε_{i j s t}, \end{array}

In Equation 1, $Y_{i j s t}$ is a test score for student i who was taught by teacher j at school s in year t. $χ_{i j s}$ is a vector that includes basic demographic and socioeconomic information of student i who was taught by teacher j at school s. The term $γ_{j s}$ is a vector of teacher fixed effects. This study includes squared and cubic terms of lagged achievement to more accurately represent nonlinear relationships between lagged achievement and current achievement (Chetty et al., 2011; Goldhaber, Walch, & Gabele, 2014).

In addition, the empirical Bayesian shrinkage method is used to address small student–teacher ratios in the data set, which, as discussed above, can increase the uncertainty of the estimates. By shrinking their value-added estimate to meet the average estimate for the entire sample, Bayesian shrinkage reduces the probability that teachers with fewer-than-average students will be misclassified. This approach is recommended by numerous researchers to decrease teacher misclassification, particularly when the student–teacher ratio is too small to compute value-added estimates (Herrmann, Walsh, Isenberg, & Resch, 2013). Moreover, although the approach does not fully account for selection bias, it does so efficiently for nonrandom assignments of teachers to schools, which is another advantage of using Bayesian shrinkage (Guarino et al., 2015). The study followed the Bayesian shrinkage approach of Grissom, Kalogrides, and Loeb (2015). Detailed information can be found in the Online Appendix B (available in the online version of the article).

After calculating the teacher’s value-added score and employing the Bayesian shrinkage method, the scores were matched to the appropriate students at each grade level.

Following Jacob, Rockoff, Taylor, Lindy and Rosen (2016), to estimate value-added scores that were not affected by the students assigned to a given teacher, the study calculated teachers’ value-added scores by using the scores of other students in the classroom while excluding the target student’s score. Most studies calculate teachers’ value-added scores by using their previous students’ performance, which helps ensure exogenous estimates; however, the scope of the LSAY study does not include information regarding teachers’ prior students’ performance. Therefore, the value-added score of one teacher with respect to a given student is here estimated based on every other student in the classroom, with the target student being omitted from the equation. Although this approach has its own limitations, it nonetheless helps address and reduce threats related to endogeneity that may occur when estimating the value-added scores. The findings and the trends are similar to the original results.

Cumulative Teacher Qualification Index

To advance research into the association between cumulative teachers’ quality and students’ short- and long-term educational success, this study uses cumulative teacher qualification indices computed from survey data on teacher qualifications—specifically, years of teaching experience, level of education, and subject-matter expertise.

To measure how many years each student had a highly qualified teacher in secondary school, the study assigned each teacher a cumulative teacher qualification index variable. Unlike the teacher’s years of experience variable, which is a continuous variable, the other teacher qualification variables needed to be dummy coded because they were categorical variables. Thus, teachers were assigned dummy codes for their highest level of education (master’s or higher = 1; bachelor’s or lower = 0) and subject-matter expertise (major or minor in their subject = 1; major and minor in out of field = 0).¹ All three variables were then matched to the appropriate students at each grade level. Students, therefore, generated a score for each qualification indicator for each year of secondary education (7th to 12th grades). The study then summed the scores into a final cumulative value for each indicator.

Table 2 illustrates how this cumulative teacher quality index works. For example, a sample student has been taught by five mathematics teachers during her secondary education. Only three teachers have at least a master’s degree or higher, giving a cumulative level of education of 3, but all five majored in the subject they teach, giving a cumulative subject-matter expertise of 5. Similarly, the five teachers to whom the students were exposed, collectively, had 49 years of teaching experience, giving them 49 years of cumulative teaching experience. Similarly, because the sum of each teacher’s value-added scores was 42, the cumulative measure of teacher effectiveness to which the students were exposed is 42.

Table 2

Illustrative Example of Measures of Cumulative Teachers’ Quality

	Teacher’s years of experience	Teacher’s level of education	Teacher’s subject-matter expertise	Teacher’s effectiveness
Year 7	5	0	1	6
Year 8	3	1	1	8
Year 9	6	0	1	18
Year 10	15	1	1	2
Year 11	—	—	—	—
Year 12	20	1	1	8
Cumulative teachers’ quality	49	3	5	42

Control Variables

To identify the role that cumulative teachers’ quality plays in students’ educational success in an unbiased manner, the present study controls for several factors in considering the measures of its analytic models. The study includes students’ gender, race/ethnic group, socioeconomic status (SES), postsecondary aspirations, course selection, and initial test scores in mathematics. In particular, the present study accounts for the effect of course selection by controlling for whether students have taken advanced mathematics courses in high school. The previous literature has shown that taking more Advanced Placement courses significantly increases student achievement, high school graduation, and even college performance (Adelman, 2006; Shettle et al., 2007). It is particularly important to consider this enrollment as well when investigating the influence of cumulative teachers’ quality because educators teaching Advanced Placement courses tend to possess higher qualifications and complete more training than their peers (Long, Iatarola, & Conger, 2009).²

Table 3 lists the names, descriptions, and LSAY labels of the variables used to estimate the association between cumulative teachers’ quality and students’ educational success.

Table 3

Descriptions of Variables in the Study

Variable name	Description	Label
Dependent variable
Student achievement	Student’s 12th-grade mathematics achievement.Standardized the variable to have an M = 0 and SD = 1.	KMTHIRT
Bachelor’s degree completion	A dummy variable indicating whether the participant has a baccalaureate degree (0 = no baccalaureate degree, 1 = earned a baccalaureate degree)	RBACC
Independent variables
Background characteristics
Gender	Dummy variable, 1 = male, 0 = female	GENDER
Race/ethnicity	A series of dummy variables indicating African American, Hispanic, Asian, and Native American other minorities, with White as the reference category	RACE
Parents’ education level	The highest level of parent’s education, categorical:0 = Less than high school diploma1 = High school diploma2 = Some college3 = 4-year college degree4 = Advanced degree	PEDUC
College aspiration	Student’s expected highest level of education0 = High school diploma only1 = Some college2 = 4-year college degree3 = Masters4 = Doctor-Professional	SEDEX
Math attitude	Summary measure of how much the student liked mathematics in high school (5-Likert-type scale, 5 being highest)	LIKEMTH
Advanced Placement	Create a binary variable where 1 means a student has taken math-related Advanced Placement during high school and 0 means a student has not taken any math-related Advanced Placement during high school. From 9th to 12th grades, survey asked every year whether a student took math Advanced Placement in each grade.	MTH9 (9th grade)MTH10 (10th grade)MTH11 (11th grade)MTH12 (12th grade)
Cohort	Dummy variable, 1 = Cohort 1, 0 = Cohort 2	COHORT
Pretest score	For older cohort students, 10th-grade math achievement and for younger cohort students, 7th-grade math achievement were used.	GMTHIMP (10th grade)AMTHIMP (7th grade)
Teacher Qualification Indexes
Years of teaching experience	Mathematics teachers’ years of teaching experience	BG2
Level of education	Create a binary variable from teachers’ survey that indicates whether a teacher has a master’s degree or higher = 1, bachelor’s or lower = 0. Teachers were asked to check their highest degree. Information was from teacher survey questionnaire that asked teachers to check their highest degree.	BE86A (associate degree)BE87A (bachelor’s degree)BE88A (master’s degree)BE89A (specialist degree)BE90A (PhD)
Subject-matter expertise	A binary variable that indicates whether a teacher has majored or minored in the subject they teach, in field teacher = 1, out of field teacher = 0Information is from teacher first (BE87C) and second (BE87B2) major and first (BE87C1) and second minor (BE87C2) in college.	BE87B1BE87B2BE87C1BE87C2
Teacher Effectiveness Indexes
Student’s math score	Student’s math standardized score in each grade	AMTHIRT (7th grade)CMTHIRT (8th grade)EMTHIRT (9th grade)GMTHIRT (10th grade)IMTHIRT (11th grade)KMTHIRT (12th grade)

Moreover, descriptive statistics of the variables in this study are presented in Table 4. As shown in Table 4, the average of students’ mathematics achievement seems to grow as students mature. Although there is variation between grades, students’ grade level growth is 3.2, on average, for mathematics. Finally, the correlation among primary predictors can be found in Table A1 in the appendix.

Table 4

Descriptive Statistics of the Variables in the Study

Variable	Analytic sample
Variable	M	SD	Minimum	Maximum
Student achievement (mathematics)
7th grade	51.58	10.09	28.46	85.07
8th grade	54.78	10.80	24.56	88.84
9th grade	59.16	12.24	28.65	94.27
10th grade	62.05	13.03	26.5	95.26
11th grade	66.21	13.06	28.26	98.62
12th grade	67.86	13.27	28.02	100.16
Baccalaureate degree attainment	.45	.49	0	1
Gender (male)	.50	.50	0	1
Race (White)	.72	.44	0	1
SES (level of parent’s education)	1.82	1.20	0	4
Postsecondary aspiration	1.82	1.20	0	4
Math attitude	2.33	1.23	0	4
Advanced placement (mathematic)	.09	.29	0	1
Mathematics
Cumulative teacher’s effectiveness	24.79	14.06	2.55	56.30
Cumulative years of teaching experience	41.17	27.95	1	146
Cumulative level of education	1.64	1.42	0	6
Cumulative subject-matter expertise	1.91	1.43	0	6

Note. SES = socioeconomic status.

Statistical Approach

Undoubtedly, teacher quality is a key determinant of student achievement, but finding ways to identify and isolate its effect has been elusive. Unfortunately, in practice, teachers are not randomly assigned to schools, and students are not randomly assigned to teachers or schools. Thus, different sets of students contribute to the grade-level means. As a result, it can be difficult to estimate the effect of teacher quality and to make causal claims unless studies are approached with rigor and caution. Because this type of nonrandom sorting is commonplace, not adequately accounting for this matter may bias the estimate of the effect of teacher quality. The research by McCaffrey and colleagues (2003) suggests that this bias can be efficiently reduced by using a school fixed-effects model. In this model, the unmeasured aspects of the environment that may differ across schools (such as per-pupil expenditure, curriculum materials, the availability of instructional supports, or the leadership of principals) are treated as uniform at the school level (Newton, Darling-Hammond, Haertel, & Thomas, 2010). By giving schools an individual intercept, the school fixed-effects model makes it possible to control for all of the variance at the school level and thus to better estimate teacher quality (Ehrenberg, Brewer, Gamoran, & Willms, 2001; McCaffrey, Lockwood, Koretz, Louis, & Hamilton, 2004). Therefore, this study controls for the differences between schools by using a school fixed-effects model. The findings from the Hausman test (χ² = 1,113.46, p < .001) also confirm that the use of fixed effects is appropriate and efficient compared with random effects models.

This study has controlled for relevant covariates and used school fixed-effects methods; nonetheless, there are a few caveats that readers should consider. First, given its nonexperimental design, the analysis in this study is not completely free of selection bias; thus, readers are encouraged to view the relationship between teacher quality and student academic success as relational rather than causal. Second, there may be a school in which a certain group of students may more likely be assigned to classrooms that have teachers with higher or lower quality. Although this study takes the selection bias between schools into account, the study cannot sufficiently address the selection bias and sorting that occur within a school. Nonetheless, this study intends to show the repercussion that this type of cumulative inequality within a school can have for the educational success of students.

Statistical Analysis

Short-Term Achievement

To examine the relationship between cumulative teachers’ quality and students’ short-term outcomes (students’ 12th-grade standardized mathematics test score), a school fixed-effects model was used as follows:

\begin{array}{l} Y_{i s} = β_{0} + χ_{i s} β_{1} + S_{i s} β_{2} + T Q_{i s} β_{3} + A P_{i s} β_{4} \\ + C_{i s} β_{5} + T E_{i s} β_{6} + P T_{i s} β_{7} + γ_{s} + ε_{i s} . \end{array}

In Equation 2, Yisis a variable indicating student i’s 12th-grade mathematics achievement at school s, and $χ_{i s}$ is a vector that includes the basic demographic and socioeconomic information of student i. The term $S_{i s}$ indicates student i’s postsecondary aspiration, $A P_{i s}$ represents the student’s high school course selection, $C_{i s}$ represents the student’s cohorts, $P T_{i s}$ indicates student i’s initial mathematics achievement test scores in the data set (7th-grade test score for the younger cohort and 10th-grade test score for the older cohort), and $γ_{s}$ is a vector of school fixed effects. $T Q_{i s}$ represents the vector of cumulative teacher qualification indices for the student, and $T E_{i s}$ represents the Cumulative Teacher Effectiveness Index for the student. To reveal the distinctive role that each measure of teacher quality plays in student achievement, the study includes five distinct models: Model 1 examines only the Cumulative Teacher Effectiveness Index; Models 2, 3, and 4 examine each cumulative teacher qualification measure one by one while accounting for the Cumulative Teacher Effectiveness Index; and, finally, Model 5 models all of the cumulative teacher quality indices together.

Long-Term Educational Attainment

The relationship between cumulative teachers’ quality and long-term student educational success was estimated using a fixed-effects model. As the models used in the present study use dichotomous dependent variables (whether students earned a bachelor’s degree), a fixed-effects logistic regression (also known as conditional logistic regression) was the best option for estimating models that control for school fixed effects:

\begin{array}{l} L o g i t (Y_{i s}) = β_{0} + χ_{i s} β_{1} + S_{i s} β_{2} + T Q_{i s} β_{3} \\ + A P_{i s} β_{4} + C_{i s} β_{5} + T E_{i s} β_{6} \\ + P T_{i s} β_{7} + γ_{s} + ε_{i s}, \end{array}

In Equation 3, $Y_{i s}$ is a dummy variable indicating whether student i at school s obtained a bachelor’s degree. The term $χ_{i s}$ is a vector that includes the basic demographic and socioeconomic information of student i, $S_{i s}$ indicates student i’s postsecondary aspiration, $A P_{i s}$ represents the student’s high school course selection, $C_{i s}$ represents the student’s cohorts, $P T_{i s}$ indicates student i’s initial mathematics achievement test scores in the data set (7th-grade test score for the younger cohort and 10th-grade test score for the older cohort), and $γ_{s}$ is a vector of school fixed effects. $T Q_{i s}$ represents the vector of the Cumulative Teacher Qualification Index for the student, and $T E_{i s}$ represents the Cumulative Teacher Effectiveness Index for the student. Like the short-term effect of teacher quality, to understand the distinct role that teacher quality plays in students’ bachelor’s degree attainment, the study includes five distinct models: Model 6 examines only the Cumulative Teacher Effectiveness Index; Models 7, 8, and 9 examine each cumulative teacher qualification measure one by one while accounting for the Cumulative Teacher Effectiveness Index; and, finally, Model 10 models all of the cumulative teacher quality indices together.

Results

Short-Term Achievement

Confirming prior research, the study found a positive relationship between 12th-grade achievement and enrollment in Advanced Placement courses, and greater postsecondary aspirations. Notably, cumulative teachers’ effectiveness and all three cumulative teacher qualification indices were shown to have a significant positive association with students’ achievement in mathematics. Improving cumulative teachers’ qualities by 1 SD results in an increase in mathematics achievement in the average range of 0.03 to 0.30 SD (Table 5).

Table 5

Relationship Between Teacher Quality and Short-Term Achievement

	Model 1	Model 2	Model 3	Model 4	Model 5
Prior achievement^a	0.053*** (0.001)	0.053*** (0.001)	0.053*** (0.001)	0.053*** (0.001)	0.053*** (0.001)
Postsecondary aspiration	0.083*** (0.008)	0.082*** (0.008)	0.082*** (0.008)	0.079*** (0.008)	0.079*** (0.008)
SES (level of parent’s education)	−0.010 (0.008)	−0.012 (0.008)	−0.011 (0.008)	−0.013 (0.008)	−0.013 (0.008)
Race (African American)	−0.145*** (0.038)	−0.146*** (0.038)	−0.144*** (0.038)	−0.149*** (0.038)	−0.149*** (0.038)
Race (Hispanic)	−0.116** (0.038)	−0.115** (0.038)	−0.113** (0.038)	−0.111** (0.038)	−0.110** (0.038)
Race (Asian)	0.070 (0.068)	0.073 (0.068)	0.076 (0.067)	0.076 (0.067)	0.078 (0.067)
Race (Native American and Others)	−0.006 (0.114)	−0.005 (0.113)	−0.002 (0.113)	−0.002 (0.114)	−0.000 (0.113)
Gender (male)	0.072*** (0.018)	0.072*** (0.018)	0.072*** (0.018)	0.069*** (0.018)	0.069*** (0.018)
Advanced Placement	0.345*** (0.033)	0.337*** (0.033)	0.335*** (0.033)	0.329*** (0.033)	0.324*** (0.033)
Cohort	−0.054 (0.038)	−0.035 (0.038)	−0.038 (0.038)	−0.040 (0.037)	−0.032 (0.039)
Cumulative teacher’s effectiveness^a	0.303*** (0.018)	0.285*** (0.019)	0.291*** (0.019)	0.267*** (0.020)	0.262*** (0.020)
Cumulative teacher’s years of experience^a		0.039** (0.014)			0.008 (0.017)
Cumulative teacher’s level of education^a			0.032* (0.013)		0.013 (0.015)
Cumulative teacher’s subject-matter expertise (major)^a				0.067*** (0.014)	0.061*** (0.015)
Constant	−3.402*** (0.052)	−3.392*** (0.052)	−3.399*** (0.052)	−3.375*** (0.052)	−3.375*** (0.052)
Adjusted R²	.648	.716	.716	.718	.718

Note. Standard errors in parentheses. SES = socioeconomic status.

Z-standardized variables.

p < .05. **p < .01. ***p < .001.

In mathematics, even after accounting for teachers’ qualification measures, cumulative teachers’ effectiveness was shown to have a positive association with students’ 12th-grade mathematics achievement. In fact, every SD increment in cumulative teachers’ effectiveness was shown to increase students’ 12th-grade mathematics achievement of 0.303 SD (Model 1). Moreover, accounting for cumulative teachers’ years of experience, a 1 SD increase in cumulative teachers’ effectiveness was positively related to an increase in students’ 12th-grade mathematics achievement of 0.285 SD. Taking cumulative teachers’ level of education into account, a 1 SD increase in cumulative teachers’ effectiveness was positively related to an increase in students’ 12th-grade mathematics achievement of 0.291 SD. Similarly, controlling for cumulative teachers’ subject-matter expertise, a 1 SD increase in cumulative teachers’ effectiveness was positively related to an increase in students’ 12th-grade mathematics achievement of 0.267 SD (see Models 2, 3, and 4). Although coefficients of cumulative teachers’ effectiveness reduced slightly after accounting for each cumulative teacher qualification measures, it still shows a positive relationship with an increase in students’ mathematics achievement. Accounting for all three cumulative qualification measures in the model, a 1 SD increase in cumulative teachers’ effectiveness was positively associated with an increase in students’ mathematics achievement of 0.262 SD (see Model 5).

Similarly, the models suggest that holding all other variables constant (including cumulative teachers’ effectiveness) a 1 SD increase in cumulative teachers’ years of experience, cumulative teachers’ level of education, or cumulative teachers’ subject-matter expertise was positively associated with students’ average mathematics achievement increases of 0.039 SD, 0.032 SD, and 0.067 SD, respectively (see Models 2, 3, and 4). Among the teacher qualification measures, cumulative teachers’ subject-matter expertise was the only measure shown to have a positive association with students’ mathematics achievement (β = 0.061) when accounting for all four proxies of cumulative teachers’ quality (see Model 5).

Long-Term Educational Attainment

As with this study’s short-term achievement findings, the cumulative teacher quality indices were found to have an overall positive relationship with students’ odds of college graduation (Table 6).

Table 6

Relationship Between Cumulative Teachers’ Quality and Long-Term Educational Attainment

	Model 6	Model 7	Model 8	Model 9	Model 10
Prior achievement^a	1.034*** (0.007)	1.033*** (0.006)	1.034*** (0.007)	1.033*** (0.007)	1.032*** (0.007)
Postsecondary aspiration	2.105*** (0.088)	2.096*** (0.088)	2.099*** (0.088)	2.084*** (0.087)	2.083*** (0.087)
SES (level of parent’s education)	1.377*** (0.054)	1.371*** (0.054)	1.374*** (0.054)	1.368*** (0.054)	1.367*** (0.054)
Race (African American)	0.957 (0.193)	0.953 (0.193)	0.959 (0.194)	0.934 (0.190)	0.936 (0.192)
Race (Hispanic)	0.671 (0.137)	0.669 (0.137)	0.676 (0.138)	0.674 (0.138)	0.672 (0.138)
Race (Asian)	1.442 (0.511)	1.446 (0.509)	1.468 (0.523)	1.493 (0.538)	1.485 (0.535)
Race (Native American and Others)	0.950 (0.468)	0.954 (0.472)	0.967 (0.479)	0.966 (0.483)	0.964 (0.485)
Gender (male)	0.804* (0.077)	0.802* (0.077)	0.803* (0.077)	0.788* (0.077)	0.789* (0.077)
Advanced Placement	2.691*** (0.523)	2.542*** (0.499)	2.597*** (0.516)	2.501*** (0.481)	2.461*** (0.484)
Cohort	1.423 (0.265)	1.557* (0.300)	1.489* (0.275)	1.523* (0.288)	1.577* (0.300)
Cumulative teacher’s effectiveness^a	1.561*** (0.160)	1.430*** (0.146)	1.504*** (0.165)	1.368** (0.148)	1.336** (0.146)
Cumulative teacher’s years of experience^a		1.206** (0.085)			1.108 (0.102)
Cumulative teacher’s level of education^a			1.098 (0.080)		0.988 (0.091)
Cumulative teacher’s subject-matter expertise (major)^a				1.301*** (0.091)	1.254** (0.101)
R ²	.280	.282	.281	.283	.285

Note. Standard errors in parentheses. SES = socioeconomic status.

Z-standardized variables.

p < .05. **p < .01. ***p < .001.

In mathematics, holding cumulative teachers’ effectiveness constant, odds of graduation were positively and significantly related to the cumulative teacher qualification indices (see Models 7 and 9). In particular, accounting for cumulative teachers’ effectiveness, a 1 SD increase in cumulative teachers’ experience, on average, multiplies the odds of obtaining a bachelor’s degree by 1.206. Similarly, even after considering cumulative teachers’ effectiveness, a 1 SD increase in cumulative teachers’ subject-matter expertise, on average, multiplies the odds of attaining a bachelor’s degree by 1.301. Moreover, even after accounting for all other cumulative teacher-quality variables, cumulative teachers’ subject-matter expertise was still shown to be a stronger predictor than cumulative teachers’ effectiveness—a 1 SD increment increase the probability of a student obtaining a bachelor’s degree by 1.254 (see Model 10).

Aligning with research by Chetty et al. (2013a, 2013b) on the enduring effect of teacher quality, cumulative teachers’ effectiveness was found to be positively and significantly associated with students’ bachelor’s degree attainment, even after controlling for all three cumulative teacher qualification measures. Specifically, a 1 SD increase in cumulative teachers’ effectiveness was shown to increase students’ chance of obtaining a bachelor’s degree by 1.561 (see Model 6). Moreover, holding cumulative teachers’ experience constant, a 1 SD increase in cumulative teachers’ effectiveness was shown to significantly multiply the odds of students’ bachelor’s degree attainment by 1.430. In addition, accounting for cumulative teachers’ level of education, a 1 SD increase in cumulative teachers’ effectiveness multiplies the odds of obtaining a bachelor’s degree by 1.504. Furthermore, holding teachers’ subject-matter expertise constant, a 1 SD increase in cumulative teachers’ effectiveness multiplies the odds of students’ bachelor’s degree attainment by 1.368 (see Models 7, 8, and 9). As with the short-term achievement results, the coefficients of cumulative teachers’ effectiveness diminished modestly while accounting for each cumulative teacher qualification measure. Nonetheless, holding all three cumulative qualification measures constant in the same model, a 1 SD increase in cumulative teachers’ effectiveness was shown to multiply the probability of students’ obtaining a bachelor’s degree by 1.336 (see Model 10).

Among the proxies of teacher quality used in this study, cumulative teachers’ level of education in mathematics appears to be the only proxy that was insignificant in students’ long-term educational success. This aspect is further reviewed in the “Discussion and Implications” section.

Discussion and Implications

This study reveals that being taught by a succession of high-quality teachers (as represented by years of experience, level of education, subject-matter expertise, and effectiveness) significantly increases the probability that a secondary school student will not only have higher achievement but also obtain a bachelor’s degree. Moreover, the study’s findings suggest that cumulative teachers’ quality as experienced by the student may be a complementary approach to better understand and interpret the genuine and enduring effect teachers have on the educational success of their students (Chetty et al., 2013b; Sanders & Rivers, 1996). Therefore, although this study’s findings concerning this topic is relational, the paper has important implications for policies and practices related to teacher quality and educational equity.

First, education research tends to treat teacher quality as a characteristic of teachers and infrequently as a student experience. Specifically, research has heavily focused on whether the quality of a single teacher and the specific quality characteristics of that teacher have significant relationships with student achievement—rather than evaluate the cumulative teachers’ quality to which students are exposed and its influence on their performance. This common approach inhibits one’s ability to explore and understand the genuine effect of teacher quality on students’ academic performance. It is in this context that this study provides new and relevant information concerning how we should approach and evaluate teacher quality. This study suggests that rather than place a single teacher under the microscope, we must turn our attention to the cumulative and sequential teacher quality experience to which students are exposed. Moreover, educational researchers and policymakers must seriously consider the likelihood that measuring the combined effects of many teachers, over many years, may be a more effective means for understanding teacher quality and effectiveness, particularly in secondary education. It is important to note that the scope of this study does not suggest that we should move away from identifying individual teacher effectiveness or holding individual teachers to account but to investigate the benefits of assessing teacher quality in a cumulative manner.

Second, the findings of this research further advocate for the need for balance between teacher qualification and teacher effectiveness when measuring teacher quality. The study does not intend to recommend that we should either return to teacher qualification measures or focus only on teacher effectiveness when measuring teacher quality. Rather, it argues that neither can be used in isolation to measure a given teacher’s quality to identify students’ educational success. As this study shows, each proxy can claim its own contribution to secondary student’s academic performance and bachelor’s degree attainment; thus, policies must consider ways to foster and staff schools with teachers who have these qualities. The study’s findings also show that teacher qualification measures can play a key role in improving the educational success of secondary students even after controlling for teacher effectiveness. Although the literature indicates almost no improvement in student achievement after the first few years of teaching, this study documents that the cumulative teachers’ years of experience has a positive and significant association with educational success. Therefore, the findings suggest that, despite a mixed record for predicting student performance, teacher qualification measures are meaningful from a cumulative perspective, showing a positive association with short- and long-term secondary students’ education success. Similarly, although the concerns raised by critics of VAM are often warranted, students who are taught by a consecutive number of high value-added teachers are more likely to succeed in the short and long term, compared with those who missed this experience. Hence, instead of measuring teacher quality using a single proxy or measuring one’s quality using cross-sectional data, educational leaders and researchers may consider measuring teacher quality more comprehensively, which includes a balance between teacher qualification and teacher effectiveness. Moreover, stakeholders in education should not only aim to hire teachers with higher qualifications and effectiveness but should also consider ways to foster higher qualification and effectiveness among teachers who are presently teaching our students.

Third, although the study does not directly focus on sorting and tracking issues, its findings are based on a school fixed-effects model. The implication is that the differences in the importance of cumulative teachers’ quality that are being discussed here are based on students’ educational experience within schools. In other words, within a school, the students in this data set are exposed to various levels of teacher quality, which, in turn, influences their short- and long-term educational performance. The previous literature has documented how sorting students by ethnicity, SES, language proficiency, or achievement can have repercussive effects on students’ educational experience across and within schools (Conger, 2005; Lucas & Berends, 2002). This also aligns with a study by Kalogrides and Loeb (2013) in which the authors state that “Sorting within schools is smaller than sorting among schools but a non-trivial amount of within school sorting occurs, particularly at the middle and high school levels” (p. 1). Therefore, to resolve and understand the inequities that are prevalent in education, we must address not only the teacher quality issues across schools and districts but also pay attention to the inequitable distribution of teacher quality occurring within schools cumulatively—an important topic that has been overlooked in the literature. Although this study has taken the number of math credits and whether students have taken Advanced Placement courses into account, due to the scope of the data it was difficult to account adequately for sorting and tracking issues within a school. As students are more likely to take different levels of math throughout their high schools, and high-performing, highly qualified teachers may gravitate or be assigned to the more advanced classes (Kalogrides & Loeb, 2013), future research needs to pay attention to the sorting and tracking issue alongside with cumulative teachers’ quality. Because disadvantaged students in the same school are more likely to be taught by more than their share of lower qualified and ineffective teachers compared with their more privileged peers (Kalogrides & Loeb, 2013), the study’s findings raise important and interesting new avenues for research related to teacher sorting and equity issues in education.

In addition, cumulative teachers’ subject-matter expertise was the only qualification measure shown to have a positive and significant relationship with students’ short- and long-term educational success. This was the case even after considering all of the model’s cumulative teacher quality indices. In other words, being taught by teachers who majored or minored in their teaching subject not only has a significant positive relationship with students’ short-term achievement but also has an enduring relationship with those students’ long-term educational success. It may seem obvious that it is preferable for a student to be taught by a teacher with subject-matter expertise. However, we must reconcile this notion with the fact that, in practice, many students are not exposed to such teachers. As many as one-in-three classrooms in most high-poverty and minority-concentrated schools tend to have teachers teaching out-of-field and a greater proportion of students who—compared with their more advantaged peers—are low performing and less likely to enroll in and graduate from 4-year colleges (Aud et al., 2010; Clotfelter, Ladd, Vigdor, & Wheeler, 2006; Lankford, Loeb, & Wyckoff, 2002). In this sense, this study reveals an opportunity to address educational problems in schools with student populations that have historically seen low achievements and low college degree attainment. By staffing our schools with teachers with subject-matter expertise, schools may be able to expand the high-school-to-college pipeline significantly. Moreover, considering that this information is readily available for school leaders to review during the hiring process, teachers’ subject-matter expertise may be a ready policy lever and play a vital role when determining the effectiveness of one’s teaching.

Interestingly, the study found the relationship between mathematics teachers’ advanced degrees and students’ college completion to be insignificant. One possible explanation is that unlike years of teaching experience and subject-matter expertise, the impact of a teacher’s advanced degree can be difficult to isolate. For example, a teacher may pursue a master’s degree program in educational leadership to prepare for an administrative position, a course of study that may develop administrative and leadership skills but not significantly alter instructional practice in the classroom. Moreover, although there are graduate-level programs that help teachers develop instructional skills or expand their pedagogy, there are also graduate-level teacher preparation programs that help students become first-time teachers. In this case, it might be presumptuous to conclude that teachers who went to graduate-level teacher preparation programs would be significantly better prepared than those who went to a program at the undergraduate level. Unfortunately, it is difficult to incorporate this type of information regarding teachers’ advanced degrees because very few of our current national longitudinal data sets collects detailed information about teachers’ levels of education. Thus, any conclusions regarding the role that teachers’ advanced degrees play in students’ long-term educational success are premature. Similarly, even if the study incorporated a degree by major information, it would still be difficult to isolate the contribution of an advanced degree because the major may have been at the undergraduate, graduate, or even at both. This complexity may be one reason why this study has, along with other studies, found inconsistent findings regarding the effectiveness of advanced degrees (Ladd & Sorensen, 2015). Future research could, therefore, incorporate a variety of information sources about teachers’ advanced degree attainment to shed further light on this matter.

Limitations and Recommendation for Future Research

Despite its contributions to our understanding of teacher quality, this study has some limitations that may be addressed by future research.

First, it has been more than three decades since the LSAY began collecting data. This aspect raises questions regarding the generalizability and applicability of the findings to the current education system, which must be weighed against the distinct advantages that the data offer for addressing the research questions explored in the study. Prior studies fail to take this cumulative aspect into account because most national representative data sets, such as the Educational Longitudinal Study of 2002 and the High School Longitudinal Study of 2009, do not collect information on students and teachers on an annual basis, rendering it difficult to observe the authentic contribution of cumulative teachers’ quality. Therefore, although the data used in this study are 30 years old, they are also a rare example of a nationally representative sample that is collected on an annual basis, which is a critical prerequisite for any examination of the complex phenomenon of cumulative teachers’ quality. There are few, if any, data sets that provide the same depth and breadth of information of this type.

With respect to the applicability of the data to teachers in a more contemporary context, according to data from the Schools and Staffing Survey (SASS), the number of teachers holding an advanced degree (i.e., master’s or higher) has increased only marginally from 47% in 1987–1988 to 56% in 2011–2012. Furthermore, over the last two decades, the average number of years of teaching experience has also changed only slightly, according to the National Center for Education (Feistritzer, Griffin, & Linnajarvi, 2011). These facts allay some—though not all—of the concerns about the age of the data.

Because many current policies, including teacher hiring, salary, and evaluation, are based on teacher quality, the data offer a longer term—yet still relevant—perspective. In other words, although the data were collected some time ago, current policies reflect similar values and incorporate similar factors, and many of the same measures of teacher quality still touch substantial portions of education budgets at every level.

In addition, although the measures of teacher quality in this study are widely used, they are not necessarily informative about what occurs in the classroom—specifically, teachers’ instructional practices and classroom management. What increases the challenge is the fact that what constitutes effective instruction varies substantially across subject and context (Brophy & Good, 1986; Lewis et al., 1999). However, research has documented that teachers’ value-added, though imperfect, can help identify effective and ineffective teachers (Chetty et al., 2010, 2013b; Goe, 2007b), helping us to investigate and understand how effective teachers act in the classroom and the common mistakes that ineffective teachers make repeatedly. Moreover, combined with classroom observations and student surveys, the MET project suggests that teachers’ value-added measures can help identify effective teaching and identify teachers who need targeted professional development (Kane & Cantrell, 2013). Nonetheless, more research is necessary to untangle the relationship between teacher quality and teaching quality and to advance our understanding of the two-shape student learning.

It is important to note that, because research about cumulative teachers’ quality has gained little attention, various conceptual, methodological, and practical concerns can be raised. Because cumulative teachers’ quality was previously an under-researched area, the goal of this study was to provide initial empirical evidence of the efficacy and legitimacy of the cumulative approach for our students’ short- and long-term education.

However, the following issues arise: (a) whether there is a difference in students’ performance if a student has been exposed to high-quality teachers early in their schooling compared with later in their education, (b) how to handle a student who has been taught by one mathematics teacher throughout her schooling, and (c) how to isolate the contribution of a single teacher’s quality in a co-teaching classroom. These issues and many other related questions are valid and, in turn, must be explored and answered in future research to better understand the effect of the sequential experience of teacher quality holistically.

Indeed, measuring students’ actual cumulative teachers’ quality experience may pose questions and raise threats to validity. Nonetheless, it should not hinder our efforts to pursue and measure students’ actual teacher quality experiences precisely and appropriately. To help improve the educational success of all students, we should pivot the conversation and consideration of an individual teacher’s quality, which is closely and strongly relevant to accountability, to the student’s cumulative exposure to teacher quality. This provides not only a better understanding of an individual student’s educational experience but also provides deeper insights into the association between schooling and the students’ education. Thus, future research should examine the questions cited above and other relevant questions regarding the cumulative approach to help us understand, measure, and evaluate the contribution of teacher quality.

Supplemental Material

EEPA769379_Supplementary_Material_CLN – Supplemental material for Pulling Back the Curtain: Revealing the Cumulative Importance of High-Performing, Highly Qualified Teachers on Students’ Educational Outcome

Supplemental material, EEPA769379_Supplementary_Material_CLN for Pulling Back the Curtain: Revealing the Cumulative Importance of High-Performing, Highly Qualified Teachers on Students’ Educational Outcome by Se Woong Lee in Educational Evaluation and Policy Analysis

Footnotes

Appendix

Table A1

Correlation Table of Primary Variables

	Cumulative teachers’ effectiveness	Cumulative years of teaching experience	Cumulative subject-matter expertise	Cumulative level of education
Cumulative teachers’ effectiveness	1.000
Cumulative years of teaching experience	0.656	1.000
Cumulative subject-matter expertise	0.686	0.656	1.000
Cumulative level of education	0.579	0.714	0.560	1.000

Acknowledgements

I am grateful to Peter T. Goff for his valuable comments and suggestions on this article. I thank the editor and anonymous reviewers for their guidance and greatly appreciate the American Educational Research Association for their generous support of this paper. I gratefully acknowledge Geoffrey Borman, Carolyn Kelley, Eric Camburn, and Geoff Mamerow for their contribution on earlier version of this manuscript.

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.

Funding

The author(s) disclosed receipt of the following financial support for the research and/or authorship of this article: This research was supported by a grant from the American Educational Research Association which receives funds for its AERA Grants Program from the National Science Foundation under Grant #DRL-0941014. Opinions reflect those of the author and do not necessarily reflect those of the granting agencies.

Notes

Author

SE WOONG LEE is an assistant professor of education in the Department of Educational Leadership and Policy Analysis at University of Missouri at Columbia. His research focuses on educator quality, educator evaluation, and the educator labor market, with an emphasis on social inequality.

References

Aaronson

Barrow

Sander

(2007). Teachers and student achievement in the Chicago public high schools. Journal of Labor Economics, 25, 95−135.

Adelman

(2006). The toolbox revisited: Paths to degree completion from high school through college. Washington, DC: U.S. Department of Education. Retrieved from https://files.eric.ed.gov/fulltext/ED490195.pdf

Archer

(1998, February 18). Students’ fortunes rest with assigned teacher. Education Week. Retrieved from https://www.edweek.org/ew/articles/1998/02/18/23dallas.h17.html

Aud

Hussar

Planty

Snyder

Bianco

Fox

M. A.

. . . Drake

(2010). The Condition of Education 2010 (NCES 2010-028). Washington DC: National Center for Education Statistics. Retrieved from https://nces.ed.gov/pubs2010/2010028.pdf

Baker

E. L.

Barton

P. E.

Darling-Hammond

Haertel

Ladd

H. F.

Linn

R. L.

Shepard

(2010). Problems with the use of student test scores to evaluate teachers. Washington, DC: Economic Policy Institute. Retrieved from https://files.eric.ed.gov/fulltext/ED516803.pdf

Betts

J. R.

Shkolnik

J. L.

(2000). The effects of ability grouping on student achievement and resource allocation in secondary schools. Economics of Education Review, 19, 1–15.

Betts

J. R.

Zau

Rice

(2003). Determinants of student achievement: New evidence from San Diego. San Francisco, CA: Public Policy Institutes of California.

Borman

G. D.

Kimball

S. M.

(2005). Teacher quality and educational equality: Do teachers with higher standards-based evaluation ratings close student achievement gaps? The Elementary School Journal, 106, 3–20.

Brophy

Good

T. L.

(1986). Teacher Behavior and Student Achievement. In Wittrock

M. C.

(Ed.), Handbook of research on teaching (pp. 328–375). London, England: Collier Macmillan.

10.

Carlin

B. P.

Louis

T. A.

(2000). Bayes and empirical Bayes methods for data analysis (2nd ed.). Boca Raton, FL: CRC Press.

11.

Chetty

Friedman

J. N.

Hilger

Saez

Schanzenbach

D. W.

Yagan

(2010). How does your kindergarten classroom affect your earnings? Evidence from Project STAR (Working Paper No. 16381). Retrieved from http://www.nber.org/papers/w16381.pdf

12.

Chetty

Friedman

J. N.

Rockoff

J. E.

(2011). The long-term impacts of teachers: Teacher value-added and student outcomes in adulthood (Working Paper No. 17699). Retrieved from http://www.nber.org/papers/w17699.pdf

13.

Chetty

Friedman

J. N.

Rockoff

J. E.

(2013a). Measuring the impacts of teachers I: Evaluating bias in teacher value-added estimates (Working Paper No. 19423). Retrieved from http://www.nber.org/papers/w19423.pdf

14.

Chetty

Friedman

J. N.

Rockoff

J. E.

(2013b). Measuring the impacts of teachers II: Teacher value-added and student outcomes in adulthood (Working Paper No. 19424). Retrieved from http://www.nber.org/papers/w19424.pdf

15.

Chingos

M. M.

Peterson

P. E.

(2011). It’s easier to pick a good teacher than to train one: Familiar and new results on the correlates of teacher effectiveness. Economics of Education Review, 30, 449–465.

16.

Clotfelter

Ladd

H. F.

Vigdor

Wheeler

(2006). High-poverty schools and the distribution of teachers and principals. North Carolina Law Review, 85, 1345–1379.

17.

Clotfelter

C. T.

Ladd

H. F.

Vigdor

J. L.

(2007). Teacher credentials and student achievement: Longitudinal analysis with student fixed effects. Economics of Education Review, 26, 673–682.

18.

Coleman

J. S.

Campbell

E. Q.

Hobson

C. J.

McPartland

Mood

A. M.

Weinfeld

F. D.

York

(1966). Equality of educational opportunity. Washington, DC: U.S. Government Printing Office. Retrieved from https://files.eric.ed.gov/fulltext/ED012275.pdf

19.

Collins

Amrein-Beardsley

(2014). Putting growth and value-added models on the map: A national overview. Teachers College Record, 116(1), 1–32.

20.

Conger

(2005). Within-school segregation in an urban school district. Educational Evaluation and Policy Analysis, 27, 225–244.

21.

Cowan

Goldhaber

(2016). National board certification and teacher effectiveness: Evidence from Washington State. Journal of Research on Educational Effectiveness, 9, 233–258.

22.

Dee

T. S.

Cohodes

S. R.

(2008). Out-of-field teachers and student achievement evidence from matched-Pairs comparisons. Public Finance Review, 36, 7–32.

23.

DiPrete

T. A.

Eirich

G. M.

(2006). Cumulative advantage as a mechanism for inequality: A review of theoretical and empirical developments. Annual Review of Sociology, 32, 271–297.

24.

Dynarski

(2016). Teachers observations have been a waste of time and money. Washington, DC: Brookings Institution. Retrieved from https://www.brookings.edu/research/teacher-observations-have-been-a-waste-of-time-and-money/

25.

Ehrenberg

R. G.

Brewer

D. J.

Gamoran

Willms

J. D.

(2001). Class size and student achievement. Psychological Science in the Public Interest, 2, 1–30.

26.

Feistritzer

C. E.

Griffin

Linnajarvi

(2011). Profile of teachers in the U.S. 2011. Washington, DC: National Center for Education Information. Retrieved from https://www.edweek.org/media/pot2011final-blog.pdf

27.

Gamoran

(2002). Beyond curriculum wars: Content and understanding in mathematics. In Loveless

(Ed.), The great curriculum debate: How should we teach reading and math? (pp. 134–162). Washington, DC: Brookings Institution Press.

28.

George

(2003). Growth in students’ attitudes about the utility of science over the middle and high school years: Evidence from the longitudinal study of American youth. Journal of Science Education and Technology, 12, 439–448.

29.

Goe

(2007). The link between teacher quality and student outcomes: A research synthesis. Washington, DC: National Comprehensive Center for Teacher Quality. Retrieved from https://files.eric.ed.gov/fulltext/ED521219.pdf

30.

Goldhaber

D. D.

(2007). Everyone’s doing it, but what does teacher testing tell us about teacher effectiveness? Journal of Human Resources, 42, 765–794.

31.

Goldhaber

D. D.

Brewer

D. J.

(1996). Evaluating the effect of teacher degree level on educational performance. In Fowler

W. J.

(Ed.), Developments in school finance (pp. 197–210). Washington, DC: National Center for Education Statistics, U.S. Department of Education.

32.

Goldhaber

D. D.

Brewer

D. J.

(1997). Why don’t schools and teachers seem to matter? Assessing the impact of unobservables on educational productivity. Journal of Human Resources, 32, 505–523.

33.

Goldhaber

D. D.

Brewer

D. J.

(2000). Does teacher certification matter? High school teacher certification status and student achievement. Educational Evaluation and Policy Analysis, 22, 129–145.

34.

Goldhaber

D. D.

Quince

Theobald

(2018). Has it always been this way? Tracing the evolution of teacher quality gaps in U.S. public schools. American Educational Research Journal, 55, 171–201.

35.

Goldhaber

D. D.

Walch

Gabele

(2014). Does the model matter? Exploring the relationship between different student achievement-based teacher assessments. Statistics and Public Policy, 1, 28–39.

36.

Graham

J. W.

Hofer

S. M.

Donaldson

S. I.

MacKinnon

D. P.

Schafer

J. L.

(1997). Analysis with missing data in prevention research. In Bryant

K. J.

Windle

West

S. G.

(Eds.), The science of prevention: Methodological advances from alcohol and substance abuse research (pp. 325−366). Washington, DC: American Psychological Association.

37.

Grissmer

D. W.

Flanagan

Kawata

J. H.

Williamson

LaTourrette

(2000). Improving student achievement: What state NAEP test scores tell us. Santa Monica, CA: RAND.

38.

Grissom

J. A.

Kalogrides

Loeb

(2015). Using student test scores to measure principal performance. Educational Evaluation and Policy Analysis, 37, 3–28.

39.

Guarino

C. M.

Maxfield

Reckase

M. D.

Thompson

P. N.

Wooldridge

J. M.

(2015). An evaluation of empirical Bayes’s estimation of value-added teacher performance measures. Journal of Educational and Behavioral Statistics, 40, 190–222.

40.

Hanushek

E. A.

Kain

J. F.

O’Brien

D. M.

Rivkin

S. G.

(2005). The market for teacher quality (Working Paper No. 11154). Retrieved from http://www.nber.org/papers/w11154.pdf

41.

Hanushek

E. A.

Rivkin

S. G.

(2006). Teacher quality. In Hanushek

E. A.

Welch

(Eds.), Handbook of the economics of education (Vol. 2, pp. 1051–1078). Amsterdam, The Netherlands: North Holland.

42.

Harris

D. N.

(2011). Value-added measures in education: What every educator needs to know. Cambridge, MA: Harvard University Press.

43.

Harris

D. N.

Sass

T. R.

(2011). Teacher training, teacher quality and student achievement. Journal of Public Economics, 95, 798–812.

44.

Haycock

(1998). Good teaching matters: How well-qualified teachers can close the gap. Thinking K-16, 3(2), 1–17.

45.

Herrmann

Walsh

Isenberg

Resch

(2013). Shrinkage of value-added estimates and characteristics of students with hard-to-predict achievement levels. Washington, DC: Mathematica Policy Research. Retrieved from https://aefpweb.org/sites/default/files/webform/Herrmann_et_al_2013_Shrinkage_and_Student_Characteristics.pdf

46.

Hong

(2010). The reciprocal relationship between parental involvement and mathematics achievement: Autoregressive cross-lagged modeling. Journal of Experimental Education, 78, 419–439.

47.

Hoxby

(2000). Peer effects in the classroom: Learning from gender and race variation (Working Paper No. 7867). Retrieved from http://www.nber.org/papers/w7867

48.

Ingersoll

R. M.

(2003). Why schools have difficulty staffing their classrooms with qualified teachers? Educational Leadership, 60(8), 30–33.

49.

Jackson

C. K.

(2012). Non-cognitive ability, test scores, and teacher quality: Evidence from 9th grade teachers in North Carolina (Working Paper No. 18624). Retrieved from http://www.nber.org/papers/w18624.pdf

50.

Jacob

B. A.

Lefgren

Sims

(2008). The persistence of teacher-induced learning gains (Working Paper No. 14065). Retrieved from http://www.nber.org/papers/w14065.pdf

51.

Jacob

B. A.

Rockoff

J. E.

Taylor

E. S.

Lindy

Rosen

(2016). Teacher applicant hiring and teacher performance: Evidence from DC public schools (Working Paper No. 22054). Retrieved from http://http://www.nber.org/papers/w22054

52.

Kalogrides

Loeb

(2013). Different teachers, different peers the magnitude of student sorting within schools. Educational Researcher, 42, 304–316.

53.

Kane

M. T.

(2017). Measurement error and bias in value-added models (ETS Research Report Series No. 17–25). Princeton, NJ: Educational Testing Services. Retrieved from https://doi.org/10.1002/ets 2.12153

54.

Kane

T. J.

Cantrell

(2013). Ensuring fair and reliable measures of effective teaching: Culminating findings from the MET Project’s three-year study. Seattle, WA: Bill and Melinda Gates Foundation. Retrieved from https://files.eric.ed.gov/fulltext/ED540958.pdf

55.

Kane

T. J.

Staiger

D. O.

(2008). Estimating teacher impacts on student achievement: An experimental evaluation (Working Paper No. 14607). Retrieved from http://www.nber.org/papers/w14607

56.

Koretz

D. M.

(2008). Measuring up: What educational testing really tells us. Cambridge, MA: Harvard University Press.

57.

Ladd

H. F.

(2008). Teacher effects: What do we know? In Duncan

Spillane

(Eds.), Teacher quality: Broadening and deepening the debate (pp. 3–26). Evanston, IL: Northwestern University. Retrieved from http://www.sesp.northwestern.edu/docs/teacher-quality.pdf

58.

Ladd

H. F.

Sorensen

L. C.

(2015). Do master’s degrees matter? Advanced degrees, career paths, and the effectiveness of teachers. Washington, DC: National Center for Analysis of Longitudinal Data in Education Research. Retrieved from https://caldercenter.org/sites/default/files/WP%20136_0.pdf

59.

Lankford

Loeb

Wyckoff

(2002). Teacher sorting and the plight of urban schools: A descriptive analysis. Educational Evaluation and Policy Analysis, 24, 37–62.

60.

Lee

S. W.

Min

Mamerow

G. P.

(2015). Pygmalion in the classroom and the home: Expectation’s role in the pipeline to STEMM. Teachers College Record, 117(9), 1–40.

61.

Lewis

Parsad

Carey

Bartfai

Farris

Smerdon

(1999). Teacher quality: A report on the preparation and qualifications of public school teachers. Washington, DC: National Centre for Educational Statistics. Retrieved from https://nces.ed.gov/pubs99/1999080.pdf

62.

Little

R. J.

(1988). A test of missing completely at random for multivariate data with missing values. Journal of the American Statistical Association, 83, 1198–1202.

63.

Little

R. J.

Rubin

D. B.

(1989). The analysis of social science data with missing values. Sociological Methods & Research, 18, 292–326.

64.

Long

M. C.

Iatarola

Conger

(2009). Explaining gaps in readiness for college-level math: The role of high school courses. Education Finance and Policy, 4, 1–33.

65.

Lucas

S. R.

Berends

(2002). Sociodemographic diversity, correlated achievement, and de facto tracking. Sociology of Education, 75, 328–348.

66.

(2001). Participation in advanced mathematics: Do expectation and influence of students, peers, teachers, and parents matter? Contemporary Educational Psychology, 26, 132–146.

67.

(2004). Determining the causal ordering between attitude toward mathematics and achievement in mathematics. American Journal of Education, 110, 256–280.

68.

McCaffrey

D. F.

Lockwood

J. R.

Koretz

Louis

T. A.

Hamilton

(2004). Models for value-added modeling of teacher effects. Journal of Educational and Behavioral Statistics, 29, 67–101.

69.

McCaffrey

D. F.

Lockwood

J. R.

Koretz

D. M.

Hamilton

L. S.

(2003). Evaluating value-added models for teacher accountability. Santa Monica, CA: RAND. Retrieved from https://www.rand.org/content/dam/rand/pubs/monographs/2004/RAND_MG158.pdf

70.

McEwan

P. J.

(2003). Peer effects on student achievement: Evidence from Chile. Economics of Education Review, 22, 131–141.

71.

Miller

J. D.

Kimmel

Hoffer

T. B.

Nelson

(2000). Longitudinal study of American youth: User’s manual. Chicago, IL: International Center for the Advancement of Scientific Literacy, Northwestern University.

72.

Miller

Roza

(2012). The sheepskin effect and student achievement: De-emphasizing the role of master’s degrees in teacher compensation. Washington, DC: Center for American Progress. Retrieved from https://files.eric.ed.gov/fulltext/ED535593.pdf

73.

Newton

X. A.

Darling-Hammond

Haertel

Thomas

(2010). Value-added modeling of teacher effectiveness: An exploration of stability across models and contexts. Education Policy Analysis Archives, 18(23), 1–23.

74.

Nye

Konstantopoulos

Hedges

L. V.

(2004). How large are teacher effects? Educational Evaluation and Policy Analysis, 26, 237–257.

75.

Papay

J. P.

(2011). Different tests, different answers the stability of teacher value-added estimates across outcome measures. American Educational Research Journal, 48, 163–193.

76.

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.

77.

Potter

Morris

D. S.

(2017). Family and schooling experiences in racial/ethnic academic achievement gaps: A cumulative perspective. Sociological Perspectives, 60, 132–167.

78.

Potter

Roksa

(2013). Accumulating advantages over time: Family experiences and social class inequality in academic achievement. Social Science Research, 42, 1018–1032.

79.

Rice

J. K.

(2003). Teacher quality: Understanding the effectiveness of teacher attributes. Washington, DC: Economic Policy Institute.

80.

Rivkin

S. G.

Hanushek

E. A.

Kain

J. F.

(2005). Teachers, schools, and academic achievement. Econometrica, 73, 417–458.

81.

Rockoff

J. E.

(2004). The impact of individual teachers on student achievement: Evidence from panel data. American Economic Review, 94, 247–252.

82.

Rothstein

(2009). Student sorting and bias in value-added estimation: Selection on observables and unobservables. Education Finance and Policy, 4, 537–571.

83.

Rothstein

(2017). Measuring the impacts of teachers: Comment. American Economic Review, 107, 1656–1684.

84.

Royston

(2005). Multiple imputation of missing values: Update of ice. Stata Journal, 5, 527–536.

85.

Rubin

D. B.

(1987). Multiple imputation for nonresponse in surveys. New York, NY: John Wiley.

86.

Sanders

W. L.

Rivers

J. C.

(1996). Cumulative and residual effects of teachers on future student academic achievement. Knoxville: University of Tennessee Value-Added Research and Assessment Center. Retrieved from https://www.heartland.org/_template-assets/documents/publications/3048.pdf

87.

Schafer

J. L.

Graham

J. W.

(2002). Missing data: Our view of the state of the art. Psychological Methods, 7, 147–177.

88.

Shettle

Roey

Mordica

Perkins

Nord

Teodorovic

. . . Kastberg

(2007). The nation’s report card: America’s high school graduates (NCES 2007–467). Washington, DC: U.S. Government Printing Office. Retrieved from https://files.eric.ed.gov/fulltext/ED495682.pdf

89.

von Hippel

P. T

. (2007). Regression with missing Ys: An improved strategy for analyzing multiply imputed data. Sociological Methodology, 37, 83–117.

90.

Wang

Wildman

(1996). The relationship between parental influences and student achievement in seventh grade mathematics. School Science and Mathematics, 96, 395–400.

91.

Weisberg

Sexton

Mulhern

Keeling

(2009). The widget effect: Our national failure to acknowledge and act on differences in teacher effectiveness. Brooklyn, NY: New Teacher Project. Retrieved from https://tntp.org/assets/documents/TheWidgetEffect_2nd_ed.pdf

92.

Willson

A. E.

Shuey

K. M.

Elder

G. H.

Jr. (2007). Cumulative advantage processes as mechanisms of inequality in life course health. American Journal of Sociology, 112, 1886–1924.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.06 MB