Preparing Students for College and Careers: The CausalRole of Algebra II

Matthew N. Gaertner • Jeongeun Kim • Stephen L. DesJardins •

Katie Larsen McClarty

Received: 10 April 2013 / Published online: 29 November 2013� Springer Science+Business Media New York 2013

Abstract In educational research and policy circles, college and career readiness is

generating great interest. States are adopting various policy initiatives, such as rigorous

curricular requirements, to increase students’ preparedness for life after high school.

Implicit in many of these initiatives is the idea that college readiness and career readiness

are essentially the same thing. This assumption has persisted, largely untested. Our paper

explores this assumption in greater depth. Using two national datasets and an instrumental

variables approach to mitigate selection bias, we evaluated the effects of completing

Algebra II in high school on subsequent college and career outcomes (i.e., persistence and

graduation as well as wages and career advancement). Results suggest Algebra II matters

more for college outcomes than career outcomes and more for students completing Algebra

II in the early 1990s than in the mid-2000s. Study limitations are discussed along with

directions for future research, such as evaluating the opportunity cost associated with

taking Algebra II for students seeking careers upon high school completion.

Keywords High school mathematics � High school course-taking � Algebra

II � College readiness � Career readiness � Instrumental variable

Introduction

Beginning with the 1983 publication of A Nation at Risk, and continuing uninterrupted in

the three decades since, American educators and policymakers have engaged in a vigorous

Electronic supplementary material The online version of this article (doi:10.1007/s11162-013-9322-7)contains supplementary material, which is available to authorized users.

M. N. Gaertner (&) � K. L. McClartyCenter for College & Career Success, Pearson, 400 Center Ridge Drive, Austin, TX 78753, USAe-mail: matthew.gaertner@pearson.com

J. Kim � S. L. DesJardinsUniversity of Michigan, 2117-C School of Education Building, 610 E. University Avenue, Ann Arbor,MI 48109-1259, USA

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debate about postsecondary readiness. The statistics presented in that congressional report

painted a rather austere picture of American preparedness for the new century: About 13 %

of all 17-year-olds in the United States were considered functionally illiterate and only

one-third could solve mathematics problems requiring several steps.

The present data offer little more encouragement: Only 76 % of high school students

currently graduate on time and there are considerable differences in graduation rates across

sub-groups. For example, the on-time high school graduation rate is below 60 % for

African-American and Hispanic students (Balfanz et al. 2012). Of the 34 developed nations

that make up the Organisation for Economic Co-operation and Development, only 8 have

lower high school graduation rates than the U.S. Moreover, even high school graduates

often lack sufficient preparation for success in postsecondary education. Bailey (2009)

estimates that approximately 60 % of community college students are referred to one or

more developmental education courses. Other evidence indicates that when graduates enter

the workforce directly after high school, they tend to be rated as deficient in mathematics,

written composition, and reading comprehension (Casner-Lotto et al. 2006).

These dismal statistics have received significant attention both in the media and among

educational policymakers, so concerted efforts are underway to improve postsecondary

preparation and career readiness among high school students. In fact, 21 states and the

District of Columbia currently require high school graduates to have completed a college-

and career-ready curriculum (Achieve 2011). In discussions of college-preparatory cur-

riculum, mathematics coursework tends to receive focused attention. Many states have

increased the minimum number of mathematics credits required to graduate from high

school, and states are also specifying particular types of mathematics courses that students

must complete (Federman 2007; Reys et al. 2007). These curricular requirements eschew

traditional tracking in high school in favor of equivalently high standards and rigorous

preparation for all students. Implied in these initiatives are the belief that college and

career readiness is a singular concept; that is, what prepares students for college also

prepares them for entry into the workforce. This assumption has persisted, largely untested.

The purpose of this paper is to explore this assumption in greater depth.

Some studies indicate that the levels of academic preparation required for colleges and

careers are essentially equivalent. For example, ACT research demonstrates that high

school students require the same level of preparation in reading and mathematics, whether

they are preparing for college or a workforce training program (ACT 2006). Similarly,

Muller and Beatty (2008) argue, ‘‘The skills that support academic success are not qual-

itatively different from those that support success in the workplace, as evidenced by the

fact that employers and post secondary institutions share concerns about the readiness of

high school graduates’’ (p. 8).

Others argue that while the knowledge and skills required for college versus careers

bear some resemblance, the need to apply those skills clearly differentiates the two paths.

Specifically, the Association for Career and Technical Education (ACTE) argues, ‘‘While

there is no debate that a rigorous level of academic proficiency, especially in math and

literacy, is essential for any post-high school endeavor, the reality is that it takes much

more to be truly considered ready for a career’’ (ACTE 2010, p. 1). Career readiness,

ACTE points out, requires the application of academic content knowledge. Additional

research points to the primacy of ‘‘soft skills’’ in distinguishing career readiness from other

forms of preparation; professionalism/work ethic, teamwork/collaboration, and oral com-

munication have been rated as the three most important skills for people entering the

workforce (Casner-Lotto et al. 2006).

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Prior studies have not explicitly addressed whether high school course-taking equally

affects college and career outcomes. Although there is a body of research on college

success, there is a dearth of studies investigating the effect of high school course-taking on

subsequent career outcomes. Research has shown enrollment in advanced coursework in

high school boosts students’ chance of success in college (Wyatt et al. 2011; Wiley et al.

2010; Goldrick-Rab et al. 2007; Perna 2004). The few studies evaluating the effect of

rigorous high school coursework on unemployment and wages provided inconsistent

results (Altonji 1992; Levine and Zimmerman 1995; Rose and Betts 2004). Furthermore,

these studies often fail to account for the self-selection of students into curricular pathways

that may result in biased estimates of the effect of course-taking on subsequent educational

outcomes (Goldrick-Rab et al. 2007).

To fill these gaps, our study compares the effect of advanced mathematics coursework

in high school on college versus career outcomes. We focus specifically on enrollment in

Algebra II, given its emphasis in educational policy.1 Additionally, Algebra II is a disci-

pline that requires students to apply critical quantitative reasoning and prepares them not

only for advanced mathematics courses but also for advanced science courses such as

chemistry and physics (ENLACE Florida 2008). Employing two National Center for

Education Statistics (NCES) databases, we investigate how completing Algebra II influ-

ences students’ college outcomes such as persistence and graduation as well as students’

career outcomes such as earnings and career advancement. The analysis of two cohorts—

the high school sophomore class of 1990 from the National Education Longitudinal Study

of 1988 (NELS; U.S. Department of Education 2000) and the sophomore class of 2002

from the Education Longitudinal Study of 2002 (ELS; U.S. Department of Education

2006)—allows us to compare the relationship between high school math course-taking and

subsequent outcomes over two time periods. Furthermore, the study employs an instru-

mental variables approach to control for the possibility that student self-selection into

Algebra II during high school may bias our results.

This rest of the article is organized as follows: First we discuss the relevant literature

regarding the relationship between high school course-taking and college and career out-

comes. Next we describe the theoretical framework and our empirical strategies. Then we

present and discuss our empirical results before concluding the paper with an acknowl-

edgment of this study’s limitations and suggestions for future research.

Literature Review

Prior research indicates that students who take more and higher levels of mathematics

courses in high school are more likely to pursue higher education and to have higher

earnings later in life (Adelman 2006; Altonji 1995; Rose and Betts 2004). Yet most

research on the relationship between high school math course-taking and subsequent

outcomes has focused on college-related outcomes or post-college career outcomes, with

much less emphasis on outcomes for students who do not attend college and instead enter

the workforce immediately following high school graduation.

Research demonstrates that students who take a more intensive secondary curriculum

are more likely to attend 4-year colleges, persist through college, and earn a bachelor’s

degree than students who take a less intensive curriculum (Adelman 1999, 2006; Choy

1 Almost half of the states in the United States currently mandate that students complete Algebra II in orderto earn a high school diploma (Achieve 2011).

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2002; Horn and Kojaku 2001). St. John and Chung’s (2006) analysis using NELS revealed

that taking advanced math courses significantly increased high school graduates’ proba-

bility of attending 4-year colleges. Adelman’s two analyses in 1999 and 2006 also suggest

a positive association between the level of math courses and degree attainment. Employing

the High School and Beyond (HSB) dataset and NELS, Adelman found that the positive

effect of math course-taking on degree attainment occurs at a different margin for the two

cohorts: while Algebra II or higher had a positive effect for the earlier cohort (HSB),

Trigonometry or above became the margin for the later cohort (NELS). Employing the

HSB data, Rose and Betts (2001) also found a significant Algebra II effect on students’

bachelor’s degree attainment. After controlling for student demographics, high school

characteristics, prior math courses, and standardized test scores, students who completed

Algebra II were 12 % more likely to earn a bachelor’s degree compared to their coun-

terparts who completed only Algebra and Geometry.

On the contrary, Bishop and Mane (2004) found that the number of academic courses

required to graduate from high school was not associated with college degree attainment

after controlling for several student-level (e.g., locus of control) and state-level (e.g.,

unemployment rates) variables. Their result suggests that the prior positive effects of math

course intensity on postsecondary outcomes may be attributable to selection, such that

students who are likely to enroll in Algebra II are also likely to complete college.

Many studies on the effect of course-taking on outcomes have been criticized for non-

random selection of students into different high school curricular paths (Long 2007). In

other words, the characteristics of students and their selection of high school courses may

not be independent of subsequent educational outcomes. Unless this correlation is ade-

quately controlled for, any non-experimental analysis of the effects of high school prep-

aration on subsequent postsecondary educational outcomes may be biased by this selection

process.

Several studies have sought to account for the selection bias inherent in students’

curricular choices by employing quasi-experimental methods such as propensity score

matching (Attewell and Domina 2008) and an instrumental variable (IV) approach (Altonji

1995). Attewell and Domina employed a propensity score matching technique that

attempts to ensure equivalence between the ‘‘treated’’ (students who took a college-pre-

paratory curriculum) and a ‘‘control’’ group (students who did not). They found that a more

demanding high school curriculum was associated with greater math and reading

achievement in high school, higher SAT test scores, and higher levels of access to and

graduation from college, compared to a less intensive curriculum. Although their findings

are consistent with prior research that documents a positive impact of rigorous course-

taking on college enrollment, they found smaller curriculum effects suggesting that studies

failing to control for selection tend to produce upwardly biased effects of academic course-

taking.

Long et al. (2012) also employed propensity score matching in examining the associ-

ations between students’ rigorous course-taking in high school and a set of secondary and

postsecondary outcomes. Using data from the Florida Department of Education, they found

that taking a high-level course in math had large positive effects on Florida Comprehensive

Assessment Test scores, high school graduation, and 4-year college enrollment. Taking

high-level math courses in high school also increased the number of credits earned, grade-

point average (GPA), and bachelor’s degree attainment.

Other studies have employed an instrumental variables approach to account for selec-

tion bias. An instrument is a variable that is conditionally correlated with the treatment

(e.g., Algebra II) but is uncorrelated with the outcome. Valid instruments can be used to

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isolate the effect of the treatment on the outcome. Altonji (1995) used the average number

of courses taken by students in an individual’s high school as an instrument when

examining the effect of an additional year of high school coursework on years of college

education and post-collegiate wages. Using the National Longitudinal Survey of Youth

(NLSY:1972), he found that an additional year of science, foreign language, and math

increased years of postsecondary education and future earnings. His results also revealed

that the effects of high-school coursework were smaller using the instrumental variables

approach than a model that inadequately accounted for selection, which is consistent with

the hypotheses that self-selection biases postsecondary outcomes estimates upward.

Meanwhile, the effect of math course-taking on occupational outcomes has generally

focused on the relationship between high school mathematics and wages. Bishop (1991)

questioned the argument that most jobs require significant competency in math and sci-

ence, and therefore high school curriculum should devote more time to core academic

subjects. Using the NLSY (79-86), he found differential effects of math competency on

wages for male and female high school graduates. Higher math competency decreased

unemployment and increased earnings for non-college bound women; the equivalent effect

was not found for males. For both males and females, however, higher scores on Algebra

and Geometry assessments significantly increased their predicted job performance and

success in job training.

Rose and Betts (2004) explained the direct effect of various high school math courses on

students’ earnings after graduation. Based on the sample from the sophomore cohort of

HSB, they found that taking Algebra and Geometry increased a student’s annual earnings

(9 years after high school graduation) by eight to nine percent. In contrast, taking voca-

tional math courses had a negative effect on annual earnings.

Altonji (1992) and Levine and Zimmerman (1995) examined the effect of preparation in

mathematics (semester hours and number of math courses, specifically) on both college

and labor market outcomes. To account for selection bias, these studies employed an

instrumental variables approach; they used the mean semester hours and number of math

courses within a high school as an instrument. Using the NLSY, Altonji (1992) found that

additional courses in mathematics had only modest direct effects on wages but significant

positive effects on wage growth rates. Meanwhile, math course-taking significantly

influenced postsecondary education attainment. Taking an extra year of math increased the

amount of education after high school by 0.34 years.

Using data from the NLSY and the HSB surveys, Levine and Zimmerman (1995) found

that taking additional math courses increased the chances of attending college. In terms of

occupational outcomes, additional courses in math increased wages for male high school

graduates. For female graduates, additional math courses predicted entry into more tech-

nical jobs and occupations that require longer job training. However, results across the two

data sets were somewhat different in terms of the sub-groups (e.g., female/male, high

school/college graduate) for whom statistically significant effects were estimated. Incon-

sistent effects across the NSLY and HSB make it difficult to draw conclusions about the

impact of mathematics curricula on education and labor market outcomes.

Generally, the studies detailed above provide evidence that math course-taking has a

positive effect on both college- and career-related outcomes, but these studies also

included several limitations. Some did employ quasi-experimental methods in an effort to

examine the causal effects of high school course-taking on college and labor market

outcomes. However, the instruments used in some of these studies (e.g., Altonji 1995;

Levine and Zimmerman 1995) may not be optimal. For example, Altonji’s instrument (the

mean semester credit hours of math within a high school) is likely to be conditionally

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correlated with unobserved characteristics that influence the outcomes, because students

from affluent schools have stronger postsecondary outcomes and are likely to take more

and higher-level academic coursework.

Though there is some evidence to suggest that high school course-taking patterns have a

positive effect on subsequent college and career outcomes, available studies have not

directly compared the two. It is the comparative impact of course-taking with which we are

principally concerned. In other words, our models should help us determine whether

advanced mathematics coursework in high school is as important for career success as it is

for college success. Our definition of outcomes should also provide a more comprehensive

treatment of the factors influencing college and career success than have previously been

available. Specifically, we examine more distal outcomes, such as persistence, graduation,

and job performance, which high-level mathematics curricula are intended to support

(Carnevale and Strohl 2013).

A better understanding of these dynamics will inform policymakers interested in pro-

moting college and career readiness. Our analyses may provide empirical support to the

notion that taking more advanced coursework in high school may enable both college and

career success. On the other hand, our analyses may suggest that course-taking behavior

influences career and college outcomes in different ways. If completion of Algebra II does

not boost a student’s odds of workforce success, individualized curricula focused on

mathematical applications in career fields and/or technical training may suitably replace

high-level mathematics courses.

Theoretical Framework

The impact of math course-taking on students’ college- and career-related outcomes may

operate in multiple ways; we examine two specific mechanisms. First, by completing high-

level math courses, students will likely gain academic content knowledge and critical

reasoning skills. Second, advanced mathematics coursework will likely be prized by col-

leges and employers when they evaluate high school graduates. These two mechanisms

may be explained using human capital theory (Becker 1965, 1993) and signaling theory

(Spence 1973, 2002).

Human capital theory applies microeconomic concepts to the study of the choices and

behavior of individuals and establishes the conceptual relationship between schooling,

individual productivity, and returns in the labor market (Becker 1965, 1993; Cohn and

Geske 1990; Mincer 1958; Schultz 1961). This theory rests primarily on the hypothesis that

more schooling increases the ability, productivity, and, thereby, wages of students who will

enter into the labor market (Becker 1993; Cohn and Geske 1990). For example, taking an

advanced math course may directly improve an individual’s labor productivity by

improving his or her reasoning skills, which should in turn boost occupational outcomes

upon high school graduation. Completing a high-level math course may also have an

indirect impact, making further increases in labor productivity possible by improving a

student’s chance of earning a bachelor’s degree (Rose and Betts 2001).

Still, taking high-level math courses incurs costs. Taking more advanced courses in high

school is difficult, imposing some psychological costs on students. Also, spending more

time studying than working represents an opportunity cost. In either case, students make

choices about their coursework, weighing the indirect and direct costs and benefits of

taking on more challenging academic content. Human capital theory, however, is limited in

helping us think about the consequences of self-selection into more challenging curricular

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paths. The theory assumes that individual students, as homogeneous agents in the presence

of perfectly competitive markets, simply sacrifice leisure and current income in order to

increase their knowledge and secure a higher future income (Carrillo 2003).

Signaling theory can help complement the human capital framework. Under this par-

adigm, course-taking would have no direct impact on a student’s productivity. Rather, the

completion of advanced courses ‘‘signals’’ to college admissions officers or employers that

a high school graduate is capable, and therefore will be successful in college or productive

on the job (Spence 1973, 2002). For instance, if Algebra II is considered to be a critical

course that differentiates applicants, Algebra II would function as a signal to admissions

officers and employers. A signal is reliable and useful when it is related to the quality of an

applicant, but it is costly to produce (Donath 2007). Because individuals with high pro-

ductivity can acquire education at a lower cost, high-productivity individuals will acquire

and use signals more often than their low-productivity counterparts.

These tenets of signaling theory also suggest that some individuals may possess

unobservable characteristics that lead them to self-select into a treatment (e.g., Algebra II).

Unobservable characteristics may then influence subsequent education or labor market

outcomes, such as graduating from college or securing a better job (Rose and Betts 2001),

rendering causal interpretations problematic. Untangling the contribution of a rigorous

curriculum to college and occupational outcomes from the contribution of students’ innate

ability (or other possibly confounding factors) depends on how well researchers account

for the endogeneity of high school course-taking.

Although both theories explain how more rigorous course-taking in high school may

improve students’ college or career outcomes, we do not know whether and how the

benefits of advanced math coursework differ for students on the career versus college path,

nor do we know how self-selection may bias estimates of those benefits. Our study seeks to

address this gap by not only comparing the effect of Algebra II on college versus career

outcomes, but also accounting for endogeneity in high school course-taking. In the next

section, we discuss the empirical approach used to mitigate selection bias.

Methodology

Research Questions

Our analyses are guided by a central, organizing research question: Do mathematics

course-taking patterns in high school influence college and career outcomes in the same

ways? Following from the general question above, we have specified separate questions in

Table 1 to focus on specific outcomes germane to college and career success. The research

questions (RQs) in Table 1 are paired, so we can examine the relative contribution of

Algebra II completion to similar outcomes across college and career domains.

Data and Variables

Two national datasets were used in this study: NELS and ELS. Both data sets contain

nationally representative samples of students who were followed longitudinally throughout

high school and beyond. Specifically, NELS sampled grade 8 students in 1988, and fol-

lowed up with additional data collections in 1990 (grade 10), 1992 (grade 12), 1994

(2 years out of high school), and 2000 (8 years out of high school). The more recent ELS

data follow a similar longitudinal structure: Grade 10 students were sampled in 2002, and

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additional data collection waves occurred with the same group in 2004 and 2006. This

parallel structure allowed us to compare the effect of Algebra II in high school across

different cohorts. The data provide detailed information on students’ high school course-

taking in addition to a number of other academic preparation variables, demographic

variables, and a range of postsecondary and occupational outcomes of interest. We lev-

eraged this detailed longitudinal data to construct a number of models testing the influence

of Algebra II completion on students’ college and career outcomes.

The explanatory variable of substantive interest is a binary indicator of whether a

student completed Algebra II during high school. Because factors that affect students’

college and occupational outcomes vary, we specified separate models for each outcome.

All models included controls for students’ demographic characteristics (e.g., race/ethnicity,

gender, and socioeconomic status) and academic achievement (high school GPA and/or

10th grade math and reading test scores). But some models included different sets of

covariates depending on the outcome being examined. For example, career-related expe-

rience, particularly students’ work experience during high school and prestige of previous

occupation (for career advancement) were included as regressors for the career outcomes.

For the models examining college GPA beyond the freshman year, college retention, and

graduation outcomes, we also controlled for freshmen GPA.

Separating the College and Career Tracks

In our analyses, we investigated two dimensions of college and career outcomes, as

indicated by the two research questions in Table 1. Because we aim to compare the effect

of taking Algebra II on each outcome, we sampled students separately for college and

career tracks for each research question. We begin with the sub-sampling rules for college

models, where we focused on academic performance (RQ 1, cumulative GPA, which was

available in NELS only), persistence (RQ 2, remaining enrolled in college at various time

points after matriculation) and graduation (RQ 2, up to 8 years after high school com-

pletion, which was available in NELS only). Students were only included in RQ 1 models

if they were admitted to college. This allowed for the inclusion of students who applied to

college and were admitted, but did not enroll immediately after high school. Students were

only included in RQ 2 models if they enrolled in college immediately following high

school. This rule was critical for retention analyses. For example, first-year retention is

measured by a student’s postsecondary status (enrolled or not) in the fall of their second

year (1993 in NELS; 2005 in ELS).

Meanwhile, career outcomes included (RQ 1) wages for the first occupation after high

school graduation along with wage change over time, and (RQ 2) career advancement, as

defined by changes in hierarchically ordered O*NET Job Zones (U.S. Department of

Table 1 Paired research questions for college and career outcomes

Researchquestion

College outcome Career outcome

1 What is the relationship between completingAlgebra II in high school and college grades?

What is the relationship between completingAlgebra II in high school and earnings?

2 What is the relationship between completingAlgebra II in high school and persistence inand graduation from college?

What is the relationship between completingAlgebra II in high school and careeradvancement?

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Labor) between initial entry and 2 years after high school graduation. For all career

models, students were considered on the career track if they never enrolled in college

within the observation windows. The college and career groups are therefore mutually

exclusive. We made this choice to ensure comparisons between Algebra II effects on

college and career outcomes were not compromised by common students across college

and career groups.2 As we note in the Limitations section later in this paper, sub-sampling

rules were established for the sake of clarity and focus, although future research should

investigate non-traditional pathways through higher education and the workforce.

Missing Data

When data are missing on explanatory or dependent variables, biased estimates may result

if the missingness is systematic, i.e., if observations with non-missing data are not rep-

resentative of the population. Whenever feasible, we utilized predictors with low levels of

missing data. For example, we used the 10th grade achievement composite as a measure of

academic ability (missing for 1.6 % of ELS students) rather than SAT or ACT scores

(missing for 33.2 % of students in ELS), which are much more likely to be absent for

lower-performing students who do not enter college or who attend 2-year rather than

4-year universities. This strategy allowed us to retain a large proportion of the original

NELS and ELS samples without having to impute. For variables that were necessary, such

as freshmen GPA (missing for 1.4 % of NELS and 17.3 % of ELS students), we employed

regression-based imputation methods.

The Statistical Model

Given the qualitative differences between college and career outcomes, we estimated the

effects of completing Algebra II on each outcome using separate statistical models. First,

we employed binary logistic regression to examine the dichotomous outcomes (i.e.

retention, graduation, and career advancement). The general model is formally described in

(1) below:

PðYi ¼ 1Þ ¼ expðbXi þ cAi þ dZi þ eiÞ1þ expðbXi þ cAi þ dZi þ eiÞ

ð1Þ

where P(Yi = 1) represents the probability that student i will achieve a given outcome

(e.g., graduate from college, maintain or increase occupational prestige). The indicator Xi

represents Algebra II completion, where 1 = completed Algebra II or higher and

0 = otherwise. The vector Ai represents a set of demographic and financial characteristics

such as gender, race/ethnicity, and socioeconomic status. The vector Zi represents stu-

dents’ academic ability (i.e., high school GPA and/or 10th grade math and reading test

scores) and college freshmen GPA (in the college retention and graduation models).

Estimated parameters are b, c, and d, associated with X, A, and Z (respectively) and ei is a

logistically distributed error term.

2 Among students who entered the workforce immediately after high school, 22 % later attended collegeand earned a GPA. Our sub-sampling rules eliminated these students from the career track and includedthem instead in the college track. Reasonable arguments can be made for including these ‘‘late-arrivers’’ inboth the career and college tracks, but as we note in the Results section, including late-arrivers in both trackswould not substantively change our findings.

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We modeled continuous dependent variables using (2) where Yi represents the college

GPA, wages, or wage change over time for student i and the regressors are defined as

above. The parameters b, c, and d were estimated using ordinary least squares (OLS)

regression.

Yi ¼ bXi þ cAi þ dZi þ ei ð2Þ

Instrumental Variables

A ‘‘naı̈ve’’ statistical model described by Eqs. (1) and (2) assumes that the error term (e) is

independent of all the regressors (A, Z, and X). However, the Algebra II regressor (X) is

likely to be endogenously related to postsecondary outcomes given students’ non-random

self-selection into different high school curricular paths.3 Unless this endogenous rela-

tionship is addressed, the coefficient estimates associated with Algebra II completion may

be biased. To mitigate this problem we employed an instrumental variable (IV) approach

(Angrist and Krueger 2001; Stock and Trebbi 2003). An instrument is a variable that is

unrelated to the error term but, conditional on the other variables, is related to the

endogenous variable (i.e., whether a student took Algebra II). A valid instrument identifies

a source of exogenous variation and uses this variation to determine the impact of a

treatment (e.g., Algebra II) on an outcome (e.g., college and career success). Instrumental

variable estimation allows a researcher to make defensible claims about the effect of the

treatment on the outcomes (Angrist and Pischke 2009) by minimizing bias due to

endogeneity.

The instrumental variables we used were (1) the unemployment rate in the zip code

where students lived during the 10th grade and (2) whether a student was 16 or older (‘‘age

16’’) in 10th grade.4 This information was obtained from the Bureau of Labor Statistics’

Local Area Unemployment Statistics. Our selection of these instruments is based on both

conceptual and empirical rationales. From a conceptual standpoint, local labor market

conditions may affect course-taking decisions in high school and thereby affect college

enrollment and completion or labor market outcomes. For example, in a weak local labor

market, students may devote more time to study by increasing the quantity or difficulty of

the courses they take in high school.

Our selection of age 16 is based on state policies that limit the amount of time younger

students can spend working. For example, in the state of New York, students under 16 are

allowed to work only 3 h per day and a total of 18 h per week, whereas 16- and 17-year-old

students can work up to 4 h per day and 28 h per week. This type of child labor standard

exists in most states and is expected to differentially impact the amount of time students

allocate to schooling versus work. Specifically, students under 16 may allocate less time to

work and more time to study, taking more or higher-level courses in high school.

3 Tests of the null hypothesis that Algebra II is exogenous were easily rejected in all the NELS models, butfor the ELS models the null was not rejected.4 For some models both instruments were included, for others only the age 16 in 10th grade instrument wasused. This determination was based on tests of instrument redundancy available in Stata’s ivreg2 command.In some cases the unemployment instrument did not add to the model, above and beyond the age 16instrument. Inclusion of redundant instruments in over-identified models, such as those that include moreIVs (e.g., unemployment rate and age 16) than endogenous regressors (e.g., Algebra II) will tend to biaspoint estimates relative to a just identified model (e.g., one including age 16 only; see Angrist and Pischke2009, for details).

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Therefore, including age 16 allows us to exploit an exogenous influence on the different

allocation of time to work across geographic locales.

If students believe that they will be unable to obtain employment when they graduate

from high school then they will tend to allocate less time to work and more time to

schooling during high school. In this scenario students are likely to increase the quantity

and/or difficulty of the courses that they take in high school. We operationalize the

exogenous influence of availability of work by using the unemployment rate (%) in the

county where a student resides during the 10th grade (for additional detail about the use of

the age 16 and unemployment rate as IVs see Bielby et al. 2013).

A causal IV model estimates the effect of the treatment on compliers, and this estimate

is referred to as the local average treatment effect (LATE; Angrist et al. 1996; Angrist and

Pischke 2009). In this study, compliers are students who are (1) at least 16 years old, (2)

roughly equally likely to work or take Algebra II, and (3) induced to take Algebra II by an

increase in the unemployment rate.

Our instruments are evaluated via the five assumptions associated with a valid IV as

proposed by Angrist et al. (1996): (1) stable unit treatment value assumption, (2) random

assignment, (3) exclusion restriction, (4) non-zero average causal effect of the instrument

on the treatment, and (5) monotonicity. The stable unit treatment value assumption

requires that an individual student’s Algebra II completion does not influence other stu-

dents’ outcomes (i.e., spillover effects). It is unlikely that one student taking Algebra II (or

not) will impact another student’s college or career outcomes. Still, students are grouped

within high schools, and it is possible that students who do not complete Algebra II are

nonetheless affected by sharing instructors or by interactions between Algebra II

instructors and other teachers, assuming any Algebra II content is covered in lower-level

classes. Because high school curricula are hierarchically structured and teaching Algebra II

without prerequisite material would be quite difficult, the risk of violating this assumption

is low. It is also assumed that for valid IVs, the treatment is consistent across all treated

groups. It is reasonable to suspect Algebra II curricula vary from school district to school

district, so district fixed effects were incorporated to address this concern.

The second assumption of a valid IV concerns random assignment, which requires that

the distribution of the instrumental variable across individuals be comparable to what

would be the case given random assignment. This means that any individual in the sample

has an equal probability of having any level of the instrumental variable. We believe 10th

graders are highly unlikely to change their residence to obtain employment in a zip code

with a lower unemployment rate. It is also unlikely that parents will choose to move their

students when they are in 10th grade: Residential mobility decreases as children get older,

especially between counties (Long 1972). In NELS and ELS, mobility rates are relatively

low. Fifteen percent of NELS students moved between 8th and 10th grade, and in ELS,

only 3 % of students transferred schools due to family relocation. Furthermore, low cor-

relations between moving and parental occupation, unemployment status, and family

income suggest students’ likelihood of changing residence is not dictated by their back-

ground. Therefore, local labor market conditions—along with factors associated with

residential choice—vary independently of students’ course-taking, college performance,

and career outcomes.

Finally, a student’s birth year and month are determined by factors that approximate

random assignment. However, it is arguable that students who are 16 or older are sub-

stantially different from their 15-year-old counterparts. For example, parents may choose

to hold a student back in 10th grade due to poor academic performance. Prior research has

demonstrated strong relationships between a student’s age and his or her academic

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performance (Angrist and Krueger 1991). Supplementary analyses suggest that controlling

for other individual characteristics, there is no significant difference in 10th grade reading

and math composite scores between 15 and 16-year-old 10th graders in the NELS and ELS

samples, indicating that the age 16 in 10th grade instrument is randomly distributed across

the NELS and ELS samples. Finally, academic performance in high school affects college

and work outcomes (Astin and Oseguera 2005; Camara and Echternacht 2000), so we

included controls for academic performance in our models (standardized 10th grade

reading and math test scores and high school GPA).

Next, satisfying the exclusion restriction assumption ensures that an IV affects the

dependent variable (e.g., graduation from college) only through its relationship with the

endogenous independent variable of interest (e.g., Algebra II). To satisfy this assumption,

the instrument must not be correlated with the error terms in Eqs. (1) and (2). The zip-code-

level unemployment rate instrument satisfies this assumption in two ways. We use

unemployment rates when students are in the 10th grade. It is arguable that the local

unemployment rate changes over time when students are in high school, affects students’

and parents’ financial status and ability to save, and thus affects students’ decisions about

pursuing college or a career after high school. However, financial conditions for high

school students tend to be stable. Local unemployment rates when students are in 10th

grade and 12th grade are highly correlated (r = 0.90). Stable annual national unemploy-

ment rates in 1990–1992 (5.6, 6.8, and 7.5 %, respectively) and 2002–2004 (5.8, 6.0, and

5.5, respectively) also suggest labor markets did not change dramatically during NELS and

ELS students’ high school careers. Finally, parents’ employment status and occupation, as

well as income level in 10th grade and 12th grade are also highly correlated (r = 0.99).

For models in which we included both the age 16 and unemployment IVs, we conducted

over-identification tests which provide evidence that the second-stage residuals are

uncorrelated with the IVs. The test assumes that the first IV (zip-code-level unemployment

rates) is properly excluded from second-stage equation. Thus, satisfying this assumption

means failing to reject the null hypothesis that instruments are correctly excluded from the

estimation of a dependent variable. In all the over-identified models we estimated (i.e.,

college outcomes for ELS students), the Sargan–Hansen v2 test failed to reject the null,5

suggesting that this assumption is met.

The fourth assumption is non-zero average causal effect of the instrument on the

treatment. This assumption requires that there be a strong relationship between the

instrumental variable and the endogenous independent variable. To address this assump-

tion, we examined a set of test statistics including the first stage F-statistic and partial-R2

values (Bound et al. 1995). Generally, our first-stage F-statistics exceeded the often used

rule of thumb of 10 and the instruments accounted for between 0.3 and 1.4 % of the total

variation in the dependent variables. Other formal tests represent improvements over these

statistics (see Baum 2008; Angrist and Pischke 2009) and are more appropriate for judging

whether there is a weak instrument problem. For example, often the Cragg-Donald Wald

F-statistic test is employed. However, this test assumes independent and identically dis-

tributed (i.i.d.) standard errors and is invalid when employing cluster-robust standard errors

as was the case in our models. Thus, we employed the Kleibergen–Paap (K–P) test, which

provides evidence about the weak instrument issue when the i.i.d. assumption is violated

(see Baum 2008, for details). Because there are no formal critical values for the K–P test, a

rule of thumb often used is 10. In every case except one (the wage change model in NELS)

5 For college GPA, Sargan v2 (1) = 0.134 (p = 0.714), and for graduation, Sargan v2 (1) = 469(p = 0.494).

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the K–P test rejects the null hypothesis that our model is unidentified, mitigating any bias

that can be introduced by weak instruments.6

Finally, monotonicity assumes the IV has a unidirectional effect on the endogenous

variable. That is, the relationship between Algebra II and our instruments should be either

positive or negative for all individuals in the sample. In other words, increases in the

unemployment rate should never result in decreases in math course taking, and students

who are 16 in 10th grade should always be less likely to take Algebra II. Our instrument is

unlikely to satisfy this assumption fully, for two reasons. First, there may be students who

choose not to take Algebra II when unemployment rates are soaring because they feel they

need to devote more time to searching for work. Second, there may be students older than

16 who are more likely to complete Algebra II than their younger counterparts. However,

this set of students represents a small fraction of our sample, and the presence of these

‘‘defiers’’ would simply place an upper bound on our estimate of the treatment effect

(Angrist and Pischke 2009; Porter 2012). Defiers account for a small portion of our sample.

In addition, in the ELS sample the proportion of students who completed or did not

complete Algebra II is stable across unemployment rates. At the lowest unemployment rate

quartile, 76 % of students completed Algebra II, whereas 69 % completed Algebra II at the

highest unemployment rate quartile. In NELS, 16-year-olds who completed Algebra II

were roughly 10 % of the sample. Moreover, for defiers, the estimated relationship

between the instrument and the endogenous variable is expected to be in the opposite

direction to that of compliers (i.e., among defiers we observe a positive relationship

between employment rates or age 16 and enrolling in Algebra II). Mathematically, if we

were to combine the estimated effects for each individual, the opposing signs would simply

push the average effects of the instruments toward zero. As long as compliers outnumber

defiers in our sample, we will be able to obtain at least a lower-bound estimate of the causal

effect of mathematics course taking on postsecondary outcomes.

The IV method discussed above is implemented in two stages. In Stage 1 we regressed a

binary variable Algebra II (Ti) on a full set of covariates (Ci) as well as the age 16 indicator

and, if warranted via the redundancy test, the 10th grade zip-code-level unemployment rate

(Vi) using a linear probability model specification.

Ti ¼ UCi þ hV i þ xi ð3ÞFrom our estimates of (3) we calculated and stored the residuals (bxi).

In Stage 2 we estimated a regression model where the outcomes were the same as in

equations (1) and (2), and the regressors were a full set of demographic and academic

controls (vectors A and Z), along with the residuals (bxi) from the Stage 1 Eq. (3) that

account for the endogeneity of Algebra II. Correcting for endogeneity using this ‘‘control

function’’ approach should result in the estimation of b (see Eqs. 1 and 2) that more

accurately approximates the causal influence of T on Y than when employing the naı̈ve

statistical model that does not account for self-selection.7,8

6 For both NELS and ELS models, results from endogeneity, over-identification, weak ID, and first-stageF-tests are included in the Online Resource.7 We also bootstrapped the entire two-stage process to account for the estimation of the residuals in stageone and address uncertainty introduced by this estimation process. For example, estimating the two-stagestructure in a single iteration would not account for these sources of uncertainty and may lead to under-estimates of the standard errors and overestimates of the significance of the regressors, potentially leading toincorrect inferences about their effects on our outcomes.8 We chose the control function approach because the method of control functions (CF) is more robust thanalternatives, such as using propensity scores (predicted values only) as controls in the outcome equation. The

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Results

Our findings speak to not only the comparative effects of Algebra II completion on college

and career outcomes, but also the persistence of those effects as students mature, and the

stability of those effects across the decades spanned by our datasets. For each research

question, before presenting final model estimates we provide basic descriptive statistics

comparing outcomes for those who completed Algebra II and those who did not.

Descriptive statistics represent raw estimates of the effects of Algebra II completion,

absent any statistical controls or correction for selection bias via instrumental variables.

We subsequently discuss estimates from final models, which incorporate both statistical

controls and Algebra II instruments and thereby offer less biased estimates of the causal

role of Algebra II in college and career outcomes. We intentionally present final models’

results in a simple and straightforward format, including only the estimated Algebra II

effects and associated levels of statistical significance. More detailed information such as

standard errors and model-specific sample sizes are included in the Online Resource.

It is also important to note up front that we observed a large increase in Algebra II

participation between NELS (1992) and ELS (2004). Overall, the percentage of students

completing Algebra II grew from 45 % in NELS to 70 % in ELS. This change in course-

taking behavior will be important to keep in mind as Algebra II effects are compared across

datasets. We discuss the implications of increases in Algebra II participation in the Dis-

cussion and Conclusions section.

Research Question 1: College GPA and Earnings

Table 2 presents sample sizes along with mean cumulative college GPAs, initial earnings,9

and earnings change over time10 for students who completed Algebra II and students who

did not. Although wage data are available in both NELS and ELS, ELS does not yet

contain college grades. As such, mean GPAs are presented only for NELS students.

The descriptive statistics in Table 2 reveal rather unsurprising patterns. Those who

complete Algebra II in high school have higher mean college GPAs than those who do not.

The relationship between Algebra II completion and earnings in the workforce is a bit less

clear-cut. While mean initial earnings are fairly similar regardless of Algebra II completion

status, earnings seem higher later in life for those who completed Algebra II.

Final model estimates, subject to statistical controls and instrumental variables methods,

are presented in Table 3. These estimates are based on linear regressions, so the numbers in

Table 3 may be interpreted as the change in the outcome (e.g., $954 less in initial annual

earnings for ELS students) attributable to completing Algebra II.

Footnote 8 continuedCF approach allows any unobservables affecting the dependent variable (Y) to be dependent on X (AlgebraII) while controlling for both the covariates (Z) and any instrumental variables. Importantly, the CFapproach explicitly models this dependence whereas alternative methods fail to incorporate such depen-dence (see Heckman and Vytlacil 2004, for details).9 In ELS, hourly, weekly, or monthly wages were rescaled to annual earnings. In NELS, reported earningsover the six months that immediately followed high school were multiplied by two.10 Wage change metrics span a longer time period in NELS than in ELS. Specifically, ‘‘initial wage’’ inNELS was collected in 1992, while ‘‘current wage’’ was collected in 2000. By contrast, ‘‘initial wage’’ inELS was collected in 2004, while ‘‘current wage’’ was collected in 2006.

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Table 3 suggests that with respect to college outcomes, finishing Algebra II in high

school pays dividends. For NELS students, Algebra II completion increased students’

cumulative college GPA by 0.83 points on average. This finding is not isolated to

cumulative GPA; Algebra II completion also increased first-year GPAs by 0.7 points (see

the Online Resource for additional details). This incremental increase across years actually

suggests Algebra II completion has a stronger impact on outcomes measured later in

college. Still, GPA estimates are derived solely from NELS, so future research may

investigate whether or not this pattern holds in more recent data.

With respect to earnings, our results are less conclusive, although Algebra II effects do

seem stronger in NELS than in ELS. To wit, Algebra II completion had a negative impact

on initial earnings11 and a significant positive impact on earnings change over time in

NELS, but no significant effect on earnings (either initial or over time) in ELS. It may

seem unorthodox to present career-related results that are largely inconclusive, but these

estimates underscore a salient point: Algebra II completion is an important determinant of

college performance, but it appears to wield a less consistent influence over career

outcomes.

Table 2 Descriptive statistics: college GPA and earnings

Outcome NELS ELS

Allstudents

CompletedAlgebra II

Did notcompleteAlgebra II

Allstudents

CompletedAlgebra II

Did notcompleteAlgebra II

College GPA

N 6,723 4,535 2,188 N/A

Mean 2.71 2.82 2.47

Initial earnings

N 2,417 615 1,802 4,054 2,074 1,980

Mean $10,211 $9,932 $10,306 $18,127 $18,462 $17,776

Earnings change

N 2,289 590 1,699 2,625 1,287 1,338

Mean $16,350 $18,417 $15,632 $3,962 $6,111 $1,895

Table 3 Algebra II effects: college GPA and earnings

Outcome Algebra II effect (NELS) Algebra II effect (ELS)

Cumulative college GPA 0.83*** N/A

Initial annual earnings -$3,255 -$954

Annual earnings change $9,452* -$6,397

* p \ 0.05; ** p \ 0.01; *** p \ 0.001

11 Initial earnings results are similar whether or not late-arrivers (students who enter the workforceimmediately after high school but eventually enroll in college and earn a GPA) are included in careermodels. If late-arrivers are included in the career track in NELS, the estimated Algebra II effect changesfrom -$3,255 to -$3,478.

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It also seems reasonable to think distal outcomes are less sensitive to Algebra II

completion than outcomes measured closer to high school exit. Thus far the results have

not provided a clear answer to that question. The next section explores some additional

distal outcomes, focusing specifically on college retention, graduation, and career

advancement.

Research Question 2: Retention, Graduation, and Occupational Prestige

Table 4 provides retention12 rates and graduation rates13 for those NELS and ELS students

who completed Algebra II and those who did not. In addition, we provide data on career

advancement—the percentage of students on the career path who maintain or increase

occupational prestige over time. Occupational prestige is operationalized via O*NET Job

Zones14 from the U.S. Department of Labor.

Table 4 suggests patterns similar to those described in the previous section. In general,

students who have completed Algebra II are more likely to stay enrolled in college and

ultimately graduate. In both NELS and ELS, first-year and second-year retention rates are

substantially higher for students who completed Algebra II. The college graduation rate in

NELS is 29.4 % points higher for students who completed Algebra II, relative to students

who did not. Comparative results on the career side are less clear-cut. The likelihood of

career advancement was slightly higher for NELS students who completed Algebra II, but

in ELS, career advancement does not vary by Algebra II completion.

In Table 5 we present Algebra II effects on retention, graduation, and occupational

prestige adjusted via Algebra II instruments and statistical controls. The estimates in

Table 5 are derived from binary logistic regression models, and the effects are expressed as

marginal probabilities. So, for example, NELS respondents who completed Algebra II are

about six percent more likely to maintain their occupational prestige over time, relative to

their counterparts who did not complete Algebra II.

Three themes are illustrated in Table 5: Algebra II completion has (1) a strong positive

effect on college outcomes, (2) a much smaller impact on career outcomes, and (3) a

weaker effect in ELS relative to NELS. For example, NELS students who complete

Algebra II were about 29 % more likely to remain in college into their sophomore year,

compared to students who did not complete Algebra II. Among ELS students the Algebra

II effect on first-year retention is still positive, but much smaller (about 8 % more likely).

Later on in a student’s college career, Algebra II completion increases a student’s chance

of graduating by 56 %. Conversely, although taking Algebra II may have had a minor

positive impact on career advancement for NELS students who entered the workforce after

high school, neither effect (in NELS or ELS) is statistically significant.

12 First-year retention is measured by a student’s postsecondary status (enrolled or not) in the fall of theirsecond year (1993 in NELS; 2005 in ELS). Second-year retention focuses on enrollment in the beginning ofa student’s third year.13 ELS data extend only 2 years beyond high school exit, so college graduation is reported only for NELSstudents.14 O*NET Job Zones are not reported explicitly in NELS. Rather, separate NCES-specific job categorieswere matched to the ELS dataset, where O*NET job categories are provided. To impute O*NET Job Zonesin NELS, we chose the modal Job Zone in ELS associated with the job categories both datasets have incommon. To test the impact of this imputation, observed O*NET Job Zones in ELS were converted toimputed Job Zones using this modal-mapping technique. Occupational prestige models in ELS were re-estimated, and Algebra II effects remained stable.

158 Res High Educ (2014) 55:143–165

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Res High Educ (2014) 55:143–165 159

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A fourth theme emerged through analysis of college retention and graduation. It seems

Algebra II completion had an incrementally increasing positive impact on more distal

college outcomes. Note that in NELS, the Algebra II effect on graduation is larger than the

effect on second-year retention, which in turn is larger than the effect on first-year

retention. Our college GPA analysis revealed a similar pattern—Algebra II completion

exhibited a stronger positive influence on college grades in later years. We suggest caution

in interpreting these trends because they are only evident among the NELS cohort. It may

be the case that Algebra II equips students with the skills required to excel later in their

college careers. However, it seems safest to view this pattern as further evidence of the

differences in Algebra II effects between NELS and ELS, rather than proof that the benefits

of advanced mathematics course-taking grow larger as time passes.

The Importance of Adjustments via Instrumental Variables Methods

Estimating and implementing instrumental variables were essential steps in this analysis,

required to mitigate non-random assignment of students to math courses. In general, the

‘‘naı̈ve’’ regression model underestimates the effect of Algebra II on subsequent outcomes

in NELS, and overestimates it in ELS, compared to the control function model. Space

constraints will not accommodate the discussion of each estimate here, but a brief sum-

mary may suffice. In NELS, the coefficient for college graduation changed from 0.167 to

0.560, and the wage change effect increased from $1,926 to $9,452. In ELS, the coefficient

for first-year retention changed from 0.111 to 0.081, and the career advancement coeffi-

cient changed from -0.039 to -0.066; the effects also became insignificant in the control

function model. Because our instruments are shown to be valid and necessary steps for

minimizing endogeneity due to non-random assignment, the results using instrumental

variables may be interpreted as less biased estimates of the impact of completing Algebra

II. Thus, instrumental variables methods may be useful for educational researchers seeking

robust estimates of the causal relationships between course-taking in high school and

outcomes along the college and career paths. Of course, some readers may be interested in

the associations (and not necessarily the causal relationships) between Algebra II and

postsecondary outcomes. To that end, parameter estimates for every model we estimated—

both with and without IVs—are provided in the Online Resource.

Discussion and Conclusions

In this section we discuss not only the implications of our findings, but also important

limitations of this study and avenues for future research. The section is parsed according to

those three topics.

Table 5 Algebra II effects: retention, graduation, and occupational prestige

Outcome Algebra II effect (NELS) Algebra II effect (ELS)

First-year retention 0.290*** 0.081

Second-year retention 0.498*** 0.204

College graduation 0.560*** N/A

Maintaining occupational prestige 0.064 -0.066

* p \ 0.05; ** p \ 0.01; *** p \ 0.001

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Summary and Implications

The analyses described above represent our attempt to address an unanswered question in

scholarship on course-taking behavior: Does completing Algebra II in high school affect

college and career outcomes in the same ways? Our results suggest the simplest answer is

‘‘No.’’ Algebra II completion is generally a stronger determinant of college success than

career success. This is true in the aggregate (most college outcomes are influenced sig-

nificantly by Algebra II completion, but few career outcomes are similarly affected), and it

is true for paired comparisons (e.g., college retention versus career advancement). This

study’s principal strength is convergence—multiple statistical models, across outcomes

both near-term and distal, revealing roughly the same lesson. If our conclusions were

founded on a single model, a single outcome, or a single dataset, one could reasonably

argue these conclusions are spurious. However, the convergence of our findings across

analysis contexts provides a ‘‘preponderance of evidence’’ suggesting college and career

readiness are not the same thing.

These findings also converge with prior research on the subject. For the most part,

extant research addresses the college-side questions we posed. For example, a variety of

studies suggest positive college outcomes are associated with advanced mathematics

course-taking in high school (Wyatt et al. 2011; Wiley et al. 2010; Goldrick-Rab et al.

2007; Perna 2004), and our results echo those researchers’ conclusions. On the career side,

we found weaker Algebra II effects, which supports the notion that college and career

readiness encompass at least somewhat different skill sets (Casner-Lotto et al. 2006). Our

results seem to contradict the findings of Carnevale and Desrochers (2003), who concluded

students who complete Algebra II in high school tend to get higher paying jobs. One

qualification is required—Carnevale and Desrochers suggested that the high-skilled

occupations experiencing the highest growth (and increased wages) may require two- or

4-year college degrees. To the extent that Algebra II completion helps students gain entry

into 2- and 4-year colleges, Algebra II completion may boost wages by way of college

degree attainment. We did not study this mechanism; rather, our focus was on career

outcomes for students who do not enroll in college and instead enter the workforce

immediately following high school. For that group, finishing Algebra II does not confer

many measurable benefits.

Our analyses also suggest that Algebra II effects were stronger in the early 1990s than in

the mid-2000s, and that finishing Algebra II may confer greater benefits later in college.

These findings suggest a variety of potential implications for educational policy and

practice (e.g., the costs and benefits of increasing Algebra II participation). To clearly

differentiate the questions we have tested empirically from the questions our findings raise,

we address those implications in further detail in the sections below.

Limitations

The limitations to this research can be grouped into two categories—the data and the focus.

First, the data: Although this study employed two commonly used, nationally represen-

tative NCES datasets, data collection waves in NELS and ELS were limited to finite time

periods. ELS, in particular, does not yet offer various college outcomes such as cumulative

GPA or graduation. This leaves a critical question unanswered: It is not possible to

examine whether the increasingly positive Algebra II effects on more distal outcomes in

NELS (i.e., cumulative GPA and graduation) replicate for ELS students. Moreover, ELS

does not contain data on more distal career outcomes, including earnings more than 2 years

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removed from high school. Given the current enthusiasm for college and career readiness

research, more recent data (e.g., from a cohort of students who finished high school in

2010) and more longitudinal data (e.g., additional data collection waves for the ELS

cohort) would be useful.

Second, the focus: Our classification rules assign students to the college or career path

based on behavior during and immediately after high school (application to and enrollment

in college, employment after high school). This strategy was necessary to keep the analysis

manageable and focused, but future course-taking research may home in on non-traditional

pathways, such as leaving college after a year and entering the workforce. More impor-

tantly, our analysis concerns Algebra II completion, and thereby may ignore other courses

(i.e., margins) that more clearly differentiate those who are college- or career-ready from

those who are not. For example, Algebra II participation rates have increased over time and

completion of the course has been enshrined in high school graduation requirements, so it

may now be the case that higher-level courses like Trigonometry or Calculus are better

signals of college readiness. Algebra II was chosen primarily to build on prior research

related specifically to that course (Kim et al. 2012). In addition, a binary predictor (i.e.,

Algebra II completion) was required because at present statistical software does not

accommodate a polytomous outcome (e.g., Algebra II versus Trigonometry versus Cal-

culus) in Stage 1 of the instrumental variables methodology. Subsequent investigations

may explore new course-taking margins in greater depth, as we detail in the next section.

Future Research and Final Thoughts

This study has aimed to fill some gaps in the postsecondary readiness literature—most

notably extending course-taking pattern research into the career space—and has unsur-

prisingly generated a host of new questions. We divide these topics into two categories:

methodological and policy-related.

From a methodological standpoint, some additional work on instrumental variables

techniques could benefit postsecondary readiness research. Specifically, as noted above,

postsecondary outcomes may be better predicted via categorical rather than binary

coursework variables. Comparing the effects of, for example, Algebra I, Geometry,

Algebra II, Trigonometry, Pre-Calculus, and Calculus completion (or some subset) on

college outcomes will help researchers more efficiently identify an appropriate mathe-

matics course-taking margin. Accomplishing that while integrating instrumental variables

methods requires advances in statistical computing, such that polytomous outcomes can be

specified at Stage 1. Such a technological advance would also support postsecondary

readiness studies in other content areas (e.g., comparing the effects of Biology, Chemistry,

and Physics).

From a policy standpoint, two avenues for future research are evident. First, the con-

sistent differences in effects estimated via ELS and NELS suggest something about

Algebra II has changed. Shifts in educational policy and Algebra II participation rates may

offer some explanation. Since the introduction of No Child Left Behind in 2001, an

increasing proportion of states have required Algebra II for high school graduation.

Accordingly, participation rates in this course have grown from 45 percent in NELS (1992)

to 70 % in ELS (2004)—a remarkable 56-% increase. Serious consequences are now

attached to failing the course. As a result, the Algebra II curriculum may have been

progressively ‘‘watered down’’ to avoid high failure rates. In short, it is possible that

Algebra II is not what it used to be. A recent Brookings report supports this logic (Loveless

2013). That analysis shows that since 1986, math scores on the National Assessment of

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Educational Progress (NAEP) have actually gone down for students who have completed

Algebra II, leading to the conclusion that the course ‘‘has lost some of its luster as a

credentialing mechanism’’ (p. 9).

Of course, as the Brookings report acknowledges, many things changed in the American

educational landscape between the early 1990s and the mid-2000s. These changes include

but are not limited to school funding policies, demographic characteristics of schools and

districts, and class sizes, so the patterns we observe cannot be taken as cause and effect. In

the future, qualitative syllabus and transcript studies may shed light on the extent to which

Algebra II curricula have changed in response to accountability pressures. New research

from the NCES and NAEP takes a meaningful first step in this direction (NCES 2013), and

suggests academic content can vary substantially within nominally identical courses (e.g.,

‘‘Geometry’’).

Finally, and perhaps most importantly, future research on course-taking in high school

should investigate the possibility of opportunity cost. Students on the career path who

enroll in and complete Algebra II in high school may divert time and resources away from

potentially more valuable pursuits, such as vocational education. This study suggests weak

Algebra II effects on some career outcomes and negative effects on others (e.g., initial

earnings). It is possible those students who took Algebra II and sought workforce entry

immediately after high school may have been better off in targeted technical or workforce

training courses. That said, we did not find any significant negative Algebra II effects on

the career side, so it seems equally possible that focusing on vocational training in high

school at the expense of high-level academic content may not improve career outcomes. In

that case, taking Algebra II will do no harm, and it will keep doors open to college for the

many students who do not solidify postsecondary plans before enrolling in high school

courses. Our analyses were not designed to answer this question, but subsequent studies

should probably consider it. The costs and benefits of separating career and college

preparation is clearly an area where more rigorous research is needed.

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