High School Mathematics Graduation Requirement and STEM-Related Outcomes
Guan Saw and Michael Broda
Michigan State University
Presented at the Association for Education Finance and Policy (AEFP) 37th Annual Conference
March 15-17, 2012
Boston, MA
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Abstract
Over the past three decades, states have steadily increased high school graduation
requirements, especially in mathematics and science. Raising curriculum standards is also a
commonly used policy instrument to advance STEM education. Yet, the influences of increased
graduation requirements on student outcomes remains to be addressed. Using the Educational
Longitudinal Study of 2002 (ELS) dataset, this study investigates the impact of math credit
requirement on several STEM-related outcomes. Results of multilevel modeling analyses
indicate that, after controlling for other student and school factors, increased math graduation
requirements have positive effects on units and levels of math course completed but negative
effects on math achievement. No significant relationships were observed among math graduation
requirements and math self-efficacy and the choice of a STEM major in college. Main and
interaction effects of these variables suggest that the linkage between math graduation
requirements and STEM-related outcomes may be more complex than previously thought.
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I. Introduction
Over the past two decades, various efforts aimed at reforming and improving science,
technology, engineering, and mathematics (STEM) education have been developed across the
country. Many advocate that increasing curriculum standards in STEM education is paramount
(e.g., National Research Council, 2011; President’s Council of Advisors on Science and
Technology, 2010; Schmidt, 2011). Following the release of the report A Nation at Risk: the
Imperative for Education Reform (U.S. Department of Education, 1983), many states started
raising high school curriculum standards, especially in mathematics and science. As of 1990,
only roughly one out of five states required the commission-recommended three course credits in
math and science (Medrich, Brown, Henke, Ross, & McArthur, 1992). By 2008, however, over
half of the states increased high school math and science credit requirements to the same
threshold while some states even have higher graduation requirements (Snyder & Dillow, 2011).
The emphasis placed on STEM curriculum standards belies a particular logic of reform
that persists today: Credit requirements affect student course-taking, course-taking influences
student achievement, and students have the potential to attain higher education and to pursue a
career in STEM-related areas if they complete more credits and higher levels of math and
science courses in high school. In other words, when states raise curriculum standards, schools
must require students to take additional math and science courses in order to graduate. If
students take more math and science classes, they learn more; as they learn more, they may
perform better in assessment test and come to believe that they can be successful. Finally, once
students realize they can succeed in math and science, they may begin to more carefully consider
pursuing a college major in a STEM-related program. This line of logic connects graduation
requirements to course-taking patterns, and course-taking to a variety of educational outcomes,
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including achievement, self-efficacy, and choice of college major. Following this theoretical
argument, one can see why policymakers have historically placed such importance on credit
requirements as a large-scale school reform lever.
Yet, empirically, the associations among the adoption of increased high school
graduation requirements in mathematics and science and students’ course-taking patterns,
achievement, and choice of STEM major remains unclear. Previous studies have made use of
older datasets, consisting of graduate cohorts in 1980s and early 1990s, to examine changes in
graduation requirements. However, after nearly twenty years, requirements have continued to
evolve: where once required three credits of math and science instead of two, now many states
are adopting standards that require four credits of mathematics and science (Carlson & Planty,
2011). This study attempts to examine the influences of these relatively recent reforms by
drawing on a newer nationally representative sample of 2004 high school graduates from the
Education Longitudinal Study of 2002 (ELS: 2002). Previously, research has tended to focus on
one outcome, usually course-taking or math achievement (e.g., Chaney, Burgdorf, & Atash,
1997; Clune & White, 1992; Schiller & Muller, 2003; Teitelbaum, 2003). Use of ELS also
allows us to examine the effects of graduation requirement on several different student STEMrelated outcomes, including math course-taking, math achievement, math self-efficacy, choice of
college major, and interrelationships among these variables within the same longitudinal dataset.
II. Development of Graduation Requirement Policies
In 1983, the National Commission on Excellence in Education released A Nation at Risk,
advocating for increasing high school course credit graduation requirements (U.S. Department of
Education, 1983). The response to the Nation report at the state level was swift: in the next ten
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years, 45 of 50 states raised high school graduation requirements (Stevenson & Schiller, 1999).
Flowing from state policy changes, most US school districts adjusted their local standards to
meet state guidelines. From 1980 to 1993, US high schools increased the amount of coursework
required in four core subjects: English, mathematics, social studies, and science (Stevenson &
Schiller, 1999). The most dramatic increase was seen in science and mathematics requirements,
which increased, on average, by about half to two-thirds of an additional year.
Since the 1990s, while graduation requirement policies continued to spread and become
more rigorous, the specific credit requirements for graduation vary substantially across states. In
their review of high school requirements over the last thirty years, Carlson & Planty (2011)
identify a trend of increased requirements in each of the last three decades (1980s, 1990s, 2000s),
beginning in the years immediately following A Nation at Risk (1983). In mathematics, for
example, in 1980, twenty-five states required one course credit to graduate, and only eleven
required two or more. By 2008, only one state required one credit, while thirty-three require two
or more. The majority of the states, 24 states in total, required three credits and there were a
dozen of states required four credits of math to obtain a high school diploma (Snyder & Dillow,
2011). Trends are similar for science, and similar, but smaller in magnitude, for English and
social studies.
III. Previous Studies
Graduation requirements and course-taking
Following the commission’s recommendations, researchers began to investigate the
impact of more rigorous academic standards on student course taking. Analyzing transcript data
from sixteen high schools in four states, Clune and White (1992) found that between 1982 and
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1988, students in the sample took significantly more (about half of a year) academic courses.
They also found that students slightly increased the level of difficulty in their coursework,
particularly in math and science. Given the researchers’ relatively small sample and limited
longitudinal perspective, they recognize that their findings are emergent, yet cautiously
optimistic about the impact of post-Nation state and local reforms.
Later research would both confirm and question elements of Clune & White’s (1992)
findings. In their analysis of 1,000 high schools participating in the National Longitudinal Study
of Schools (NLSS), Stevenson & Schiller (1999) found that an increase of one year in state-level
graduation requirements was significantly related to increases in student course taking in all four
core subjects. Using data from the 1990 National Assessment of Educational Progress (NAEP)
and the 1990 High School Transcript Study, Chaney, Burgdorf, and Atash (1997) found that
schools with high credit requirements were associated with increased math course taking. In
addition, they found this effect disproportionately benefitted low-income and minority students,
suggesting that these policies not only benefit all students, but also have more impact on specific
student subgroups.
Additional research has added more perspective on the issue of course taking.
Analyzing a nation-wide sample of 1992 public high school graduates from the National
Education Longitudinal Study (NELS) dataset, Teitelbaum (2003) found that higher graduation
requirements in mathematics and science influenced students to earn more credits in these
subjects, but found no evidence that students completed this additional coursework in advanced
classes, suggesting that schools adjusted to new standards by creating additional low- and midlevel math and science electives, rather than requiring more advanced courses for all students.
Using the same survey data, Schiller & Muller (2003) echoed Teitelbaum’s course-taking
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findings, confirming that increased requirements did result in more course-taking. However,
they note that students who are freshmen at the time of policy implementation (1988) would
eventually take more units and more rigorous math courses, suggesting that the impact of the
policy may vary significantly depending on a students’ academic status in high school. Overall,
previous research on course-taking and math graduation requirements has relied on data from
NLSS (1980, 1992), NELS (1988-1992), and NAEP (1990). More recent evidence is needed to
examine the impact of more recent graduation requirement reform on student course-taking
patterns.
Graduation requirements and educational outcomes
While the connection between graduation requirements and course taking appears fairly
clear, the subsequent linkage between course taking and achievement is more opaque.
Examining the impact of graduation requirements on math achievement, Teitelbaum (2003)
found that student test score gains did not vary by high school graduation requirement policy. In
other words, an increase in math course requirements did not appear to be related to an increase
in math achievement on standardized tests. Even when schools required more advanced
coursework, students did not appear to be learning more.
Chaney, Burgdorf, & Atash (1997) examined similar questions in their analysis of 1990
NAEP and high school transcript data. They found no overall effect of increased graduation
requirements on NAEP math achievement, and offered two possible explanations: first, most
students in the sample had already exceeded minimum requirements for math, and thus would be
unaffected by the policy change, or second, students appeared to take additional math courses
that did not advance their math achievement. This second explanation echoes Teitelbaum’s
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(2003) argument that high schools expanded their math course offerings without necessarily
increasing the number of advanced courses. However, the researchers do point out that
graduation requirement policies may have disproportionately strong positive effects for students
who they describe as “marginal in their motivation and skills” (Chaney, Burgdorf, & Atash,
1997: 229) by coercing students into taking additional advanced math courses that they
otherwise would not have chosen.
Along with math achievement, additional educational measures have recently been
investigated by research on graduation requirements. Bishop & Mane (2001), for example, found
that increased math credit requirements were associated with higher high school dropout rates
and lower rates of postsecondary enrollment, which suggests a possible burnout effect when
students are faced with more rigorous standards. Thus, research on the relationship between
graduation requirements, course-taking, and educational outcomes has uncovered tentative
negative effects, but also the possibility of positive effects for lower-performing students who
might otherwise not pursue additional math coursework. These observations lead us to examine
several additional educational outcomes, especially those STEM-related such as college major
and math self-efficacy which is a strong predictor of entering STEM careers,.
Graduation requirements, course-taking, and choice of STEM major
Beyond course taking, math achievement, and math self-efficacy, school reformers have
also advocated curriculum and graduation requirement reform as a policy instrument to
encourage new workers in the STEM professions (Bureau of Labor Statistics, 2009; Lacey &
Wright, 2009; National Science Board, 2004). To bolster the STEM workforce, recent reports
and policy initiatives (e.g., National Research Council, 2007; President’s Council of Advisors on
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Science and Technology, 2010) draw attention to the need to increase the rigor of math and
science preparation in U.S. schools and to expand the number of students who pursue advanced
degrees and careers in STEM fields. In spite of many policy efforts to advance STEM education,
few studies have been conducted to understand the factors associated with STEM outcomes.
Teitelbaum (2003), for example, did not examine the relationship between graduation
requirements and STEM outcomes, but notes this line of inquiry is greatly needed to advance our
understanding of how to expand participation in postsecondary institutions.
Current research on predicting STEM outcomes has been heavily focused on
understanding how individual characteristics (such as gender and identity), family background,
and academic ability are related to pursuing a STEM degree (e.g., Maple & Stage, 1991;
Nicholls, Wolfe, Besterfield-Sacre, & Shuman, 2010; Trusty, 2002). From a practical viewpoint,
these are not policy variables that can easily be intervened. As Coleman (1966) notes, the
variation of these characteristics within a school far outweighs the variation between schools,
posing a stiff obstacle to implementing widespread reform. Graduation requirements, however,
are policy tools that can be manipulated more readily, and could perhaps produce positive
outcomes for minority and low-performing students and schools.
Trusty’s (2002) analysis of NELS student data on course-taking provides a starting point.
Examining the relationship between high school course-taking and the eventual choice of a math
or science major in college, Trusty (2002) found that girls who took more math courses in high
school were more likely to choose a math or science-related major in college. For boys, only
physics was related to a math or science major; all other courses showed no significant
relationship. In the past decade, more recent data from ELS have made it possible to perform a
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similar investigation to determine the long-term effects of continual graduation requirement
reform on postsecondary outcomes in relation to STEM-related fields.
Research Questions and Hypotheses
We believe that the widespread increase of graduation requirements in mathematics will
result in students taking more advanced-level math courses, such as Pre-calculus, Calculus, and
Trigonometry. Despite earlier findings (Teitelbaum, 1992) that suggest requirements may have
little impact on advanced course taking, we argue that time was a significant factor in
implementation. In the early 1990s, many states had not fully adopted more rigorous
requirements, and those that did had only a few years’ experience. Now, more than twenty years
later, we argue that the full impact of the policies can be measured. Following this logic, we
would also expect to see student math achievement scores and self-efficacy measures would vary
by school requirement on math coursework. Likewise, if more students are taking more units
and advanced math, more students may be likely to choose a STEM-related major in college,
since they will be more familiar and comfortable with prerequisite courses in many STEM
programs. The considerations above led to the following research question:
RQ1: How have increased graduation requirements in mathematics impacted:
a) Students’ math course taking patterns in high school?
b) Students’ math achievement and self-efficacy in high school?
c) Students’ choice of STEM-related majors in college?
In addition, we hypothesize that increased course-taking will be related to increased math
achievement, increased math self-efficacy, and increased choice of STEM college majors. We
believe that these three measures are related, yet we hesitate to offer a preliminary theory as to
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the specific causal mechanisms. For example, students may take more courses because they
have higher self-efficacy in math, or they may develop higher self-efficacy in math by taking
more math courses. The same seems to be true of selecting a college major. Thus, we posit that
these concepts will be related, but stop short of suggesting a causal directionality. We seek to
answer the following research question:
RQ2: How are math course-taking patterns related to math achievement and
self-efficacy in high school, and choice of STEM-related majors in college?
Given the complexity of high school social and educational environments, we would
expect that student outcomes, including standardized test scores, self-beliefs, and postsecondary
attendance, would vary significantly by student subgroup. Research on STEM outcomes, for
example, has focused on a perceived gender gap in the number of women pursuing STEM
majors and careers (Nicholls, Wolfe, Besterfield-Sacre, & Shuman, 2010). Research on student
achievement (Coleman et al., 1966; Jencks & Brown, 1975; Maple & Stage, 1991) has long
indicated gaps in math achievement between students of different ethnic backgrounds and
socioeconomic statuses. We expect that all three outcomes may vary significantly between one
or more student subgroup. This study, therefore, also set out to answer the following research
question:
RQ3: Do the above relationships among math graduation requirement and
STEM-related outcomes vary by student subgroup (gender, race,
socioeconomic status, initial math ability, and math self-efficacy)?
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IV. Methodology
Data
This study uses data from the Educational Longitudinal Study of 2002 (ELS: 2002), the
most recent U.S. nationally-representative survey conducted by the National Center for
Educational Statistics (NCES). The ELS provides a large sample of high school students,
including about 15,000 respondents from 750 schools, who were followed over time from tenthgraders in 2002, with follow-ups in 2004 and to postsecondary education in 2006 (Ingels et al.,
2007). In addition to data collected from the students, the dataset also includes information from
their parents, teachers, and school administrators, as well as their high school transcripts.
For our analyses studying effects of math graduation requirements on course taking
patterns, achievement, and self-efficacy in math, we restrict our sample to respondents who
participated in the tenth and twelfth grade surveys (base year and first follow up). The size of this
restricted sample is 15,244. Based on Rubin (1987), we impute missing values of variables other
than student background characteristics and our variable of interest (e.g. math graduation
requirement) using the multiple imputation (MI) estimation method in IBM SPSS Statistics 19.
Observations with missing data on the relevant variables (student gender, race/ethnicity, SES,
and math graduation requirement in high school) are dropped, which brings the analysis sample
down to 14,118. For analyses using different outcome variables, we further restrict the analysis
sample to respondents who provided information of those outcome variables in the first or
second follow up survey. All of the analyses use the appropriate normalized student level panel
weights provided by ELS to adjust for the oversampling of certain groups while minimizing the
effects of large sample sizes on standard errors and tests of statistical significance.
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Measures
Independent Variable (School-level). The primary independent variable is years of math
coursework required by the school to meet the graduation requirement (1 = at least 1 years but
less than 3, 2 = at least 3 years but less than 4, 4 = 4 years).
Dependent Variables (Student-level). The following educational outcomes are examined
in this study: (1) units of math completed (ordinal: 1 = zero unit, 2 = >0 - 1.99, 3 = 2.0 - 2.99, 4 =
3.0 - 3.99, 5 = 4.0 - 4.99, 6 = 5.0 - 5.99, 7 = 6.0 or more), (2) levels of math course taken
(ordinal: 1 = No math course or math course is other, 2 = Pre-algebra, general or consumer math,
3 = Algebra I, 4 = Geometry, 5 = Algebra II, 6 = Trigonometry, pre-calculus, or calculus), (3)
12th grade math standardized test scores (continuous: 19.82 to 79.85), (4) math self-efficacy
(continuous: -2.039 to 1.811), (5) choice of major in first year and second year in college
(binary: 0 = non-STEM, 1 = STEM).
Student-level Explanatory Variables. In terms of individual level control variables, we
use gender (0 = male, 1 = female), race/ethnicity (0 = White/Asian, 1 = minority), socioeconomic status (range: -2.11 to 1.82), student educational expectation (1 = less than high school
graduation, 7 = obtain PhD, MD, or other advanced degree), parent educational expectation (1 =
less than high school graduation, 7 = obtain PhD, MD, or other advanced degree), 10 th grade
math standardized scores (range: 19.38 to 86.68), 10th grade math self-efficacy (range: -1.831 to
1.772), 10th grade math interest (range: 0 to 3; constructed by combining three indicators: “gets
totally absorbed in math”, “thinks math is fun”, and “mathematics is important”), and high
school program (two dummies: general education = 0, college preparatory = 1; general education
= 0, vocational education = 1).
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School-level Explanatory Variables. In our models, school-level control variables include
urbanicity (two dummies: suburb = 0, urban = 1; suburb = 0, rural = 1), school sector (two
dummies: public = 0, Catholic = 1; public = 0, private = 1), percent of free/reduced lunch price
student (1 = 0-5%, 7 = 76-100%), percent of minority students (0-100%), school mean
achievement (range: 32.76 to 67.24), and percent of graduates go to four-year college (0 = none,
6 = 75-100%).
Table 1 shows descriptive statistics for the total sample (n=14,118). The means and
standard deviations of all variables are presented by years of math coursework required by high
school (e.g. 1-2 year, 3 year, and 4 year). Approximately 23.4% of the sample reported that math
coursework to graduation required by their school were one to two years, 62.0% were three
years, and 14.6% were four years. Overall, high school students attending school with higher
requirement on math coursework tend to complete more units and higher levels of math courses,
have higher degree of self-efficacy, go to 4-year colleges and choose STEM as a major in
college. Those individuals who went to schools required four year math courses to graduation
performed better than their counterparts in math standardized test and possessed higher degree of
math self-efficacy measured in sophomore and senior year respectively. However, students who
came from high schools required three year of math coursework scored lower than those who
attended schools required two year of math coursework.
While no gender differences were observed in regards to attending schools with differing
standards of math coursework, our data show that schools with higher math course requirements
enrolled a higher proportion of students who are minority, came from wealthier families, have
higher educational aspirations, have higher educational expectations from their parents, and have
been placed in college preparatory academic tracks. When examining school characteristics by
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years of math coursework required, we see that urban, Catholic, and private schools tend to set
higher standards of graduation in terms of math coursework whereas suburb, rural, and public
schools have comparatively lower requirement. In general, high schools with lower percentages
of free/reduced lunch students, higher percentages of minority students, higher performance in
test scores, and higher percentages of graduates go to four year colleges were more likely to
require their students to complete more credits in math. The complete descriptive statistics
appear in Table 1.
[TABLE 1 ABOUT HERE: SEE APPENDIX]
Analytic Techniques
The questions of whether high school math graduation policies were associated with
students’ math course taking patterns, achievement, self-efficacy, and choice of college major in
STEM fields requires a multilevel analytic strategy. The ELS, which employs a nested sampling
frame of schools, follows students randomly selected within schools, and allows researchers to
use multilevel modeling. We were concerned not only with variation in students’ STEM-related
educational outcomes across schools with differing math coursework requirement (direct
effects), but also with whether the relationships of students’ STEM-related educational outcomes
with their social backgrounds and math ability and self-efficacy varied across schools
(interaction effects). One common statistical technique for analyzing hierarchical data (in this
case, students nested within schools) and cross-level effects is Hierarchical Liner Modeling
(HLM), which allows simultaneous consideration of factors from different levels of analysis
(Raudenbush & Bryk, 1986, 2002).
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In order to answer our research questions, we used a two-level model to identify
significant student- and school-level factors that predict math course taking patterns,
standardized test scores, self-efficacy, and choice of college major in STEM (binary variable).
Specifically for course-taking outcome variables, which are in ordinal scales, we used the
following model:
Level 1:
𝜂𝑚𝑖𝑗 = 𝛽0𝑗 + 𝛽𝑝𝑗 𝑋𝑝𝑖𝑗 + 𝐷𝑚𝑖𝑗 𝛿𝑚𝑗
Level 2:
𝛽0𝑗 = 𝛾00 +𝛾01 (𝑀𝑎𝑡ℎ 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑚𝑒𝑛𝑡)𝑗 + 𝛾0𝑞 𝑆𝑞𝑗 + 𝜇0𝑗
𝛽𝑝𝑗 = 𝛾𝑝0 for p > 0, or
for interaction effects, 𝛽𝑝𝑗 = 𝛾𝑝0 + 𝛾𝑝1 (𝑀𝑎𝑡ℎ 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑚𝑒𝑛𝑡)𝑗
where the 𝜂𝑚𝑖𝑗 in the level 1 equation is the expected log odds for individual i in school j to take
on a value of m or lower for the ordinal outcome variable with m categories. For the outcome
variable of units in math, m can take a discrete value of 1 to 7; with 1 = “zero unit”, 2 = “>0 1.99”, 3 = “2.0 - 2.99”, 4 = “3.0 - 3.99”, 5 = “4.0 - 4.99”, 6 = “5.0 - 5.99”, 7 = “6.0 or more”. As
for the outcome variable of levels of math courses, m can take a discrete value of 1 to 6; with 1 =
“no math course or math course is other”, 2 = “pre-algebra, general or consumer math”, 3 =
“algebra I”, 4 = “geometry”, 5 = “algebra II”, 6 = “trigonometry, pre-calculus, or calculus”. The
𝛽0𝑗 is the intercept, or the average log odds for m = 1 (i.e. zero unit) for the sample. The
individual level explanatory variables, including female, minority, SES, student educational
expectation, parent educational expectation, 10th grade math standardized score, 10th grade math
self-efficacy, 10th grade math interest, college preparatory program, and vocational education,
are represented by 𝑋𝑝𝑖𝑗 . The corresponding coefficients of these variables are indicated by 𝛽𝑝𝑗 .
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The last term in the equation, 𝐷𝑚𝑖𝑗 𝛿𝑚𝑗 , represents a set of thresholds (𝛿𝜇 ) which indicate the
difference between the intercepts for the first and each of the other contrasts.
At level-2, we model the level intercept using school level variables, 𝑆𝑞𝑗 : urban, rural,
Catholic, private, percent of free/reduced lunch students, percent of minority students, school
mean achievement, and percent of graduates go to 4-year college. Because we are primarily
interested in the main effects of these variables on math course taking outcomes, no attempt is
made to model level-1 slopes. Our level-2 model is a random-intercept model and the level-2
equations make clear that the level-1 intercept, or the average log odds of completing more units
or higher levels of math courses, is determined by school requirement on math and other school
factors, plus a random error 𝜇0𝑗 . The level-1 slopes of individual predictors are assumed to be
fixed.
For outcome variables of math standardized scores and self-efficacy which are in
continuous scale, we employed the following model:
Level 1:
𝑌𝑖𝑗 = 𝛽0𝑗 + 𝛽𝑝𝑗 𝑋𝑝𝑖𝑗 + 𝜀𝑖𝑗
Level 2:
𝛽0𝑗 = 𝛾00 +𝛾01 (𝑀𝑎𝑡ℎ 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑚𝑒𝑛𝑡)𝑗 + 𝛾0𝑞 𝑆𝑞𝑗 + 𝜇0𝑗
𝛽𝑝𝑗 = 𝛾𝑝0 for p > 0, or
for interaction effects, 𝛽𝑝𝑗 = 𝛾𝑝0 + 𝛾𝑝1 (𝑀𝑎𝑡ℎ 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑚𝑒𝑛𝑡)𝑗
where 𝑌𝑖𝑗 is the math standardized scores or self-efficacy of student i in school j. The intercept
for the jth school is given here as a fixed component 𝛽0 and a random component, 𝜇0 , indicates
the random effect of the school level on the outcome variable. 𝑋𝑝𝑖𝑗 is vector of covariates at the
individual level and represent student background, while 𝑆𝑞𝑗 is a vector of covariates at the
17
group level and represents school characteristics. The random term 𝜀𝑖𝑗 represents the
unexplained variation for students within a school. The level-1 slopes of individual variables,
𝛽𝑝𝑗 , are assumed to be fixed.
For outcome variables of choice of college major in STEM fields which is in
dichotomous form, we used the following model:
Level 1:
𝜂𝑖𝑗 = 𝛽0𝑗 + 𝛽𝑝𝑗 𝑋𝑝𝑖𝑗
Level 2:
𝛽0𝑗 = 𝛾00 +𝛾01 (𝑀𝑎𝑡ℎ 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑚𝑒𝑛𝑡)𝑗 + 𝛾0𝑞 𝑆𝑞𝑗 + 𝜇0𝑗
𝛽𝑝𝑗 = 𝛾𝑝0 for p > 0, or
for interaction effects, 𝛽𝑝𝑗 = 𝛾𝑝0 + 𝛾𝑝1 (𝑀𝑎𝑡ℎ 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑚𝑒𝑛𝑡)𝑗
where 𝜂𝑖𝑗 is the average log-odds of choosing STEM as a college major for individual i in school
j. Here, 𝑋𝑝𝑖𝑗 is a vector of individual level predictors that represent student background. At level2, we model the intercept 𝛽0𝑗 as a function of the school level explanatory variables, 𝑆𝑞𝑗 . We
treat the remaining level-1 slope coefficients, 𝛽𝑝𝑗 , as fixed. All models were estimated using
HLM 6.08 software (Raudenbush, Bryk, Cheong, & Congdon, 2004) based on restricted
maximum likelihood estimation method. Level-1 and level-2 predictors in all models were
centered on the grand mean produce adjusted school means.
V. Findings
Unconditional Models
Examinations of variance components for each unconditional model in our study indicate
that variations in all outcome variables (units and levels of math courses, math standardized
18
scores, math self-efficacy, and college major in STEM) exist at the school level (level 2).
Therefore, the use of HLM is appropriate in each model, given that the variance between schools
is consistently significant (p<0.001).
Effects of Math Graduation Requirement on Course Taking Patterns
Table 2 presents the estimations generated by the hierarchical ordered logit equation
regarding the differences in the units and levels of math completed by students at schools with
high- and low-graduation requirements. The positive coefficient for the high school math
graduation requirement variable suggests that students at schools with high graduation
requirements were more likely to take more (B= 0.68, p<0.001) and advanced (B= 0.31,
p<0.001) math courses than their counterparts at schools with lower graduation requirements. In
full model of level of math estimation, the significant interaction effects of minority student and
math requirement (B= 0.29, p<0.01) indicate that the effect of graduation requirement policy is
larger for non-white and non-Asian students. Our data also show that students who with lower
math ability but attended schools with higher standards in math curriculum may unable to
complete the same amount of math courses that completed by their peers (B= -0.01, p<0.1). The
other estimations of student- and school-level predictors are expected and mostly consistent with
previous literature. For example, students from high socio-economic families and in Catholic
schools tend to complete more and advanced courses (these results are available from the
authors.)
[TABLE 2 ABOUT HERE: SEE APPENDIX]
19
Effects on Math Achievement and Self-Efficacy
The previous section showed that students at schools with higher graduation requirements
completed more and advanced math coursework than their peers at schools with lower
requirements. Policymakers may believe that students who completed more and higher level
courses in math will perform better on assessment tests. The results in Table 3 indicate that
students who completed more and advanced math coursework not only scored higher in math
achievement (units in math: B= 0.49, p<0.001; level of math: B= 1.25, p<0.001) but also showed
higher levels of math self-efficacy (units in math: B= 0.07, p<0.001; level of math: B= 0.03,
p<0.05). However, our models revealed that high school graduation requirement policy is
negatively associated with student achievement in math standardized test (B= -0.38, p<0.001),
after controlling for other student- and school-level factors. The negative effect of math
requirement on academic performance is somewhat larger specifically for minority students (B=
-0.51, p<0.5). Interestingly, our model shows students who have taken more math courses
because of the school graduation requirement performed worse in math assessment test (B= 0.32, p<0.01). In terms of self-efficacy in math, although it varied by gender and prior math
ability, the outcome measure did not differ by the years of math coursework required by schools
(B= 0.01, p>0.1). However, we find that math graduation requirements have positive effects on
math self-efficacy among minority students (B= 0.12, p<0.5) and high SES students (B= 0.06,
p<0.5).
[TABLE 3 ABOUT HERE: SEE APPENDIX]
20
Effects on Choice of College Major in STEM
Table 4 reports the likelihood of a student choosing a college major in STEM-related
fields during their first and second year in college. We find no significant effects of math
graduation requirements on either first- (B= 0.06, p>0.1) or second-year (B= -0.12, p>0.1)
college students choosing a STEM major. In terms of the relationship between math course
taking patterns and math performance in high school and college major in college, our analyses
show that units in math completed by students (1st year major: B= 0.25, p<0.001; 2nd year major:
B= 0.26, p<0.001), math achievement (1st year major: B= 0.05, p<0.001; 2nd year major: B=
0.05, p<0.001), and math self-efficacy (1st year major: B= 0.16, p<0.01; 2nd year major: B= 0.17,
p<0.01) are positively associated with majoring in STEM fields, but levels of math coursework
taken by students are not related to choosing a STEM major (1st year major: B= -0.02, p>0.1; 2nd
year major: B= 0.09, p>0.1). Surprisingly, we find that minority students who attended high
school with higher requirement in math coursework were more likely to enroll in STEM-related
major in their first year in college (B= 0.38, p<0.5).
[TABLE 4 ABOUT HERE: SEE APPENDIX]
VI. Conclusion and Discussion
Our analysis of the impact of math graduation requirements on course taking,
achievement, self-efficacy, and choice of STEM major produced several findings worthy of
discussion. First, echoing the results of Chaney et al. (1997), Clune & White (1992), and
Schiller & Muller (2003), we find that generally increased graduation requirements in math are
positively associated with course-taking, both in terms of units in math, and in terms of levels
21
(Pre-Algebra, Algebra, etc.). Yet, those students who with lower initial math ability but attended
schools with higher requirements in math may unable to complete as many as math courses their
peers can. We also find a positive interaction effects between the graduation requirement and
non-White and non-Asian students on units in math and math self-efficacy, which further
supports Chaney et al. (1997) findings that increased requirements may disproportionately
benefit minority students.
Moving beyond course taking, we find that more rigorous math requirements are
negatively related to math achievement, after controlling for other student and school factors.
These findings support those found in Teitelbaum (2003), and stand in contrast to the earlier
work of Alexander, Cook, & McDill (1978) and Jencks & Brown (1975), who found positive
associations between rigorous curriculum and math achievement on the Scholastic Aptitude Test
(SAT). We argue that this discrepancy is largely due to the association of interest in earlier
studies. Prior to the 1980s, much less emphasis was placed on uniform rigorous state graduation
requirements- in 1980, 25 states required one credit of math to graduate, and 15 additional states
set no minimum requirement (Carlson & Planty, 2011). Therefore, studies conducted prior to the
1980s did not measure the impact of state graduation policies on achievement, but rather
measured the impact of rigorous coursework. As our own results show, students who may
choose to take more rigorous courses (often due to interest or high initial math self-efficacy) do
see better outcomes in achievement and postsecondary enrollment. However, the impact of the
state policy still appears to be negative when looking at all students, especially those students
who have taken more math courses because of the school graduation requirement.
Finally, our analysis indicates that math graduation requirements are not associated with
a student declaring a STEM major in their first year of college, and negatively associated with
22
declaring a STEM major in their second year. Again, we turn to the possibility of student
burnout as a potential driving force here. Perhaps students that were required to take more math
courses in high school become frustrated or tired of math, and pursue studies in other areas. Just
as with the postsecondary enrollment outcome, we find that total units of math taken in high
school is positively related to pursuing STEM fields. This suggests a similar selection effect
whereby students who are interested in STEM careers choose to take more courses regardless of
requirements, and then are also more likely to choose a college major in STEM.
This study has several limitations that deserve mention. First, the ELS dataset does not
contain prior measures for career interest and math achievement prior to tenth grade. Other
datasets, such as NELS (1988) have these measures, but are significantly older. Second, due to
limited coursework data in ELS, only math requirement was used to examine the effects of
graduation policy in this study. It would be interesting to study the impact of credit requirement
policy on STEM-related outcomes by examining some other high school subjects such as science
and technology. Finally, we caution against applying the principle of causal inference to this
study. This research design was neither experimental, nor quasi-experimental, meaning that one
should hesitate to argue that graduation requirements are the sole, or even the most significant,
predictor of these outcomes. Further research on this topic could possibly employ a quasiexperimental matched sample, which compared students with high graduation requirements with
a similar set of students with low requirements, paired according to demographics. This could
more accurately filter out possible latent variables that this approach is not able to account for.
To summarize, the impact of math graduation requirements appears to be mixed. On one
hand, increasing math requirements does encourage students to take more and advanced math
classes, a key priority that policy makers cite in changing state curricula. However, the link
23
between policy and course taking does not extend much farther. Beyond this outcome, we find
no indication that school math requirements boost math achievement, or the choice of a STEM
major in college. Students who like math and take more courses regardless of requirements see
positive effects in achievement and enrollment. All other students do not appear to benefit. These
findings suggest that the linkage between course-taking, achievement, self-efficacy, and STEM
major choice may be more complex than has been previously understood, and that increasing
graduation requirements may produce unintended and unanticipated negative consequences.
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27
APPENDIX
Table 1. Descriptive Statistics on Key Variables, by Years of Math Coursework Required
(n=14,118)
Years of Math Coursework Required
1-2 year
3 year
4 year
(n=3,305)
(n=8,748)
(n=2,065)
mean (sd)
mean (sd)
mean (sd)
Dependent variables (Level-1 outcomes)
Units in math courses completed
2.94
3.31
3.62
(1.09)
(1.08)
(1.06)
Level of math courses completed
4.85
5.09
5.34
(1.29)
(1.12)
(1.01)
12th-grade math standardized score
50.90
50.60
52.20
(10.07)
(10.06)
(10.10)
12th-grade math self-efficacy
-0.06
0.01
0.18
(0.99)
(0.99)
(0.99)
Attended 2-year college
0.30
0.25
0.20
(0.46)
(0.44)
(0.40)
Attended 4-year college
0.44
0.49
0.59
(0.50)
(0.50)
(0.49)
1st-year college major in STEM
0.16
0.17
0.19
(0.36)
(0.37)
(0.39)
2nd-year college major in STEM
0.23
0.22
0.25
(0.42)
(0.41)
(0.43)
Student background (Level-1 predictors)
Female
0.50
0.50
0.51
(0.50)
(0.50)
(0.50)
Minority
0.26
0.35
0.34
(0.44)
(0.48)
(0.48)
Socioeconomic Status
0.00
0.04
0.16
(0.72)
(0.75)
(0.75)
Student educational expectation
5.05
5.15
5.36
(1.45)
(1.44)
(1.34)
Parent educational expectation
5.26
5.41
5.53
(1.28)
(1.27)
(1.23)
10th-grade math standardized score
51.10
50.50
51.80
(9.95)
(9.90)
(10.14)
10th-grade math self-efficacy
-0.04
0.01
0.12
(0.94)
(0.95)
(0.96)
10th-grade math interest
1.41
1.41
1.43
(0.67)
(0.70)
(0.69)
General education
0.41
0.35
0.25
(0.49)
(0.48)
(0.43)
College preparatory
0.49
0.56
0.66
(0.50)
(0.50)
(0.47)
Vocational education
0.10
0.10
0.09
(0.30)
(0.30)
(0.29)
Table 1. Descriptive Statistics on Key Variables, by Years of Math Coursework Required
(n=14,118) (continued)
Years of math coursework required
2 year
3 year
4 year
(n=3,305)
(n=8,748)
(n=2,065)
mean (sd)
mean (sd)
mean (sd)
School Characteristics (Level-2 predictors)
Urban
0.27
0.34
0.42
(0.44)
(0.47)
(0.49)
Suburb
0.53
0.48
0.42
(0.50)
(0.50)
(0.49)
Rural
0.20
0.18
0.16
(0.40)
(0.39)
(0.37)
Public
0.87
0.77
0.70
(0.34)
(0.42)
(0.46)
Catholic
0.06
0.15
0.15
(0.24)
(0.35)
(0.36)
Private
0.07
0.08
0.15
(0.25)
(0.28)
(0.36)
Percent of free/reduced lunch students
3.35
3.20
2.83
(1.73)
(1.95)
(1.92)
Percent of minority students
0.26
0.35
0.34
(0.24)
(0.28)
(0.29)
School mean achievement
50.80
50.60
51.80
(4.68)
(5.03)
(5.96)
Percent of graduates go to 4-year college
4.25
4.51
4.94
(1.00)
(1.18)
(1.21)
Source: Educational Longitudinal Study of 2002 (ELS), National Center for Education Statistics, U.S.
Department of Education.
29
Table 2. Multilevel Ordered Logit Models for Course Taking Patterns in Mathematics
Fixed effect coefficients
Math requirement (MR)
Units in Mathematics
(n=12,318)
Model 1
Model 2
Model 3
Level of Math Courses
(n=12,950)
Model 1
Model 2
Model 3
0.63***
(0.07)
0.68***
(0.08)
X
X
0.23***
(0.04)
0.06
(0.06)
-0.04
(0.07)
0.12
(0.10)
0.17***
(0.04)
0.05
(0.06)
0.04***
(0.00)
-0.01†
(0.00)
0.16***
(0.03)
-0.03
(0.04)
0.34***
(0.06)
Level-1 Predictors
Level-2 Predictors
Female (ref: Male)
0.68***
(0.07)
X
X
0.22***
(0.04)
Female x MR
Minority (ref: White/Asian)
-0.04
(0.07)
Minority x MR
SES
0.17***
(0.04)
SES x MR
10th grade math scores
0.04***
(0.00)
10th grade math scores x MR
10th grade math self-efficacy
0.16***
(0.03)
10th grade math self-efficacy x MR
Intercept
Threshold 2
Threshold 3
Threshold 4
Threshold 5
Threshold 6
Random effect variances
Intercept
*** p<.001 ** p<.01 * p<.05 †p < .10
0.33***
(0.06)
X
X
0.26***
(0.04)
0.03
(0.07)
0.31***
(0.04)
0.11***
(0.00)
0.08*
(0.03)
0.31***
(0.06)
X
X
0.27***
(0.04)
0.00
(0.07)
0.03
(0.06)
0.29**
(0.11)
0.31***
(0.04)
-0.03
(0.07)
0.11***
(0.00)
-0.01
(0.01)
0.07*
(0.03)
-0.08
(0.05)
-4.49***
1.83***
4.12***
5.91***
7.44***
9.60***
-4.71***
1.90***
4.40***
6.40***
8.08***
10.32***
-4.71***
1.89***
4.39***
6.40***
8.08***
10.33***
-0.36***
1.42***
2.40***
3.29***
5.07***
-0.22***
1.95***
3.23***
4.32***
6.28***
-0.23***
1.95***
3.24***
4.33***
6.30***
0.94***
0.87***
0.88***
0.56***
0.43***
0.43***
Note: This table shows the unstandardized coefficient and standard errors (in parentheses) for fixed effect
estimates and variances for random effect estimates.
Table 3. Multilevel Models for Achievement and Self-Efficacy in Mathematics
12th Grade Mathematics
Standardized Scores
(n=11,327)
Model 1
Model 2
Model 3
Fixed effect coefficients
Math requirement (MR)
-0.36
(0.34)
Level-1 Predictors
Level-2 Predictors
Units in math
-0.37***
(0.09)
X
X
0.50***
(0.07)
Units in math x MR
Level of math courses
1.24***
(0.07)
Level of math courses x MR
Female (ref: Male)
-0.67***
(0.11)
Female x MR
Minority (ref: White/Asian)
-0.39**
(0.15)
Minority x MR
SES
0.58***
(0.08)
SES x MR
10th grade math scores
0.75***
(0.01)
10th grade math scores x MR
10th grade math self-efficacy
0.34***
(0.06)
10th grade math self-efficacy x MR
Intercept
55
50.05***
(0.21)
50.98***
(0.06)
12th Grade Mathematics
Self-Efficacy
(n=9,501)
Model 3
Model 5
Model 6
-0.38***
(0.09)
X
X
0.49***
(0.06)
-0.32**
(0.09)
1.25***
(0.07)
0.08
(0.12)
-0.67***
(0.10)
0.14
(0.15)
-0.40**
(0.14)
-0.51*
(0.20)
0.59***
(0.08)
0.19
(0.12)
0.75***
(0.01)
-0.01
(0.02)
0.36***
(0.06)
0.18*
(0.08)
0.06*
(0.03)
51.01***
(0.06)
0.02
(0.02)
0.01
(0.02)
X
X
0.07***
(0.01)
0.03*
(0.01)
-0.10***
(0.02)
-0.02
(0.03)
-0.02
(0.02)
0.02***
(0.00)
0.28***
(0.02)
0.05***
(0.01)
0.01
(0.02)
X
X
0.07***
(0.01)
0.00
(0.02)
0.03*
(0.01)
0.00
(0.02)
-0.10***
(0.02)
0.05
(0.03)
0.02**
(0.14)
0.12*
(0.05)
-0.02
(0.02)
0.06*
(0.03)
0.02***
(0.00)
-0.00
(0.00)
0.28***
(0.02)
-0.02
(0.02)
0.05***
(0.01)
Random effect variances
Intercept
19.57***
0.73***
0.69***
0.05***
0.03***
0.03***
Level-1
80.02
17.49
17.46
0.94
0.73
0.73
*** p<.001 ** p<.01 * p<.05 †p < .10
Note: This table shows the unstandardized coefficient and standard errors (in parentheses) for fixed effect
estimates and variances for random effect estimates.
31
Table 4. Multilevel Bernoulli Models for College Major in STEM versus Non-STEM
1st Year College Major - STEM
(n=6,678)
Model 1
Model 2
Model 3
Fixed effect coefficients
Math requirement (MR)
0.07
(0.07)
Level-1 Predictors
Level-2 Predictors
Units in math
0.03
(0.07)
X
X
0.25***
(0.06)
Units in math x MR
Level of math courses
-0.01
(0.08)
Level of math courses x MR
12th grade math scores
0.05***
(0.01)
12th grade math scores x MR
12th grade math self-efficacy
0.16**
(0.05)
12th grade math self-efficacy x MR
Female (ref: Male)
-1.24***
(0.09)
Female x MR
Minority (ref: White/Asian)
0.14
(0.11)
Minority x MR
SES
0.07
(0.07)
SES x MR
Intercept
Random effect variances
Intercept
*** p<.001 ** p<.01 * p<.05 †p < .10
2nd Year College Major - STEM
(n=4,613)
Model 1
Model 2
Model 3
0.06
(0.09)
X
X
0.25***
(0.05)
0.02
(0.10)
-0.02
(0.08)
-0.02
(0.15)
0.05***
(0.01)
-0.01
(0.01)
0.16**
(0.05)
0.02
(0.08)
-1.24***
(0.09)
0.09
(0.14)
0.10
(0.11)
0.38*
(0.15)
0.06
(0.07)
0.01
(0.09)
-0.10
(0.08)
-0.15†
(0.08)
X
X
0.26***
(0.06)
0.09
(0.08)
0.05***
(0.01)
0.18**
(0.05)
-0.58***
(0.09)
0.27*
(0.13)
0.02
(0.08)
-0.12
(0.09)
X
X
0.26***
(0.06)
0.05
(0.11)
0.09
(0.08)
0.08
(0.12)
0.05***
(0.01)
-0.02
(0.01)
0.17**
(0.05)
0.01
(0.08)
-0.58***
(.09)
0.03
0.13
0.25†
(0.13)
0.14
(0.18)
0.02
(0.08)
0.10
(0.12)
-1.57***
(0.04)
-1.83***
(0.05)
-1.86***
(0.05)
-1.14***
(0.04)
-1.28***
(0.05)
-1.30***
(0.05)
0.14***
0.17***
0.17***
0.14***
0.18***
0.18***
Note: This table shows the unstandardized coefficient and standard errors (in parentheses) for fixed effect
estimates and variances for random effect estimates.