Journal of Urban Mathematics Education
July 2015, Vol. 8, No. 1, pp. 31–61
©JUME. http://education.gsu.edu/JUME
KEITH E. HOWARD is an associate professor in the College of Educational Studies at Chapman Univer-
sity, One University Drive, Orange, CA, 92866; email: khoward@chapman.edu. His research interests in-
clude equity in mathematics access and instruction, technology integration into K–12 instruction, and statisti-
cal analyses of large-scale data sets.
MARTIN ROMERO is an assistant professor in the Department of Mathematics at Santa Ana College,
1530 W. 17th Street, Santa Ana, CA, 92706; email: romero_martin@sac.edu. His research interests include
equity in mathematics education, teaching for social justice, and developmental mathematics education at
post-secondary institutions.
ALLISON SCOTT is a director of research and evaluation at the Level Playing Field Institute, 2201
Broadway, Oakland, CA 94612; email allison@lpfi.org. Her research interests include examining structural
and social/psychological barriers facing underrepresented students of color in pursuing and persisting in Sci-
ence, Technology, Engineering, and Mathematics (STEM) fields, and the effectiveness of interventions and
initiatives to improve STEM outcomes among students from underrepresented backgrounds.
DERRICK SADDLER is a visiting mathematics instructor in the College of Arts and Sciences at the Uni-
versity of South Florida Sarasota-Manatee, 8350 N. Tamiami Trail C263, Sarasota, FL 34243; email: dsad-
dler@usf.edu. His research interests include curriculum effectiveness in mathematics education, organization
of mathematics content, students’ learning of algebra, and analysis of large-scale data sets.
Success after Failure: Academic Effects and
Psychological Implications of Early Universal
Algebra Policies
Keith E. Howard
Chapman University
Martin Romero
Santa Ana College
Allison Scott
Level Playing Field Institute
Derrick Saddler
University of South Florida
In this article, the authors use the High School Longitudinal Study 2009 (HSLS:09)
national database to analyze the relationships between algebra failure, subsequent
performance, motivation, and college readiness. Students who failed eighth-grade
Algebra I did not differ significantly in mathematics proficiency from those who
passed lower-level courses, but initially demonstrated significantly lower math-
ematics interest, mathematics utility, and mathematics identity. Both groups were
less likely than the general population to meet college requirements in the eleventh
grade, although students who passed a lower-level mathematics course fared better
than those who failed Algebra I. Implications for policies addressing mathematics
course enrollments are discussed.
KEYWORDS: ability grouping, educational policy, equity, longitudinal studies,
mathematics education
ublic education in the United States was ushered in with a notion that allowing
equal access to education for all children would make the American dream pos-
sible for anyone who was willing to work for it. A good education was presumed to
P
http://education.gsu.edu/JUME
mailto:khoward@chapman.edu
mailto:romero_martin@sac.edu
mailto:allison@lpfi.org
mailto:dsaddler@usf.edu
mailto:dsaddler@usf.edu
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 32
be both a mechanism for ensuring equal opportunity to upward mobility and a
means of raising the acumen and contributions of the populace at large. Horace
Mann tabbed the common school as the “great equalizer,” suggesting that its estab-
lishment would drive down poverty and crime while equipping an intelligent popu-
lace to develop the natural and material resources to benefit society as a whole
(Cremin, 1957/1979). But, if education were to operate today as an equalizer that
affords every citizen an opportunity through academics, then providing equal access
to course sequences that chart a path to college would seem to be an approach with-
out controversy. However, a long and historical debate over whether high schools
should provide the same “college preparatory” curriculum to all students, as op-
posed to a curriculum tailored to students’ individual interests, abilities, and needs,
complicates the issue (Mirel, 2006). Nowhere is this debate more evident today
than in the controversy surrounding mathematics instruction in secondary schools
in the United States.
The sequence and level of mathematics courses taken in secondary school are
critical factors that influence access to higher education. Algebra in particular has
been identified as a key factor in academic trajectory, mainly for its perceived role
as a “gatekeeper” course into higher education (National Mathematics Advisory
Panel, 2008; Walston & McCarroll, 2010). Research findings support this notion of
gatekeeper as being more than just rhetoric; when taken in eighth grade, algebra has
been shown to be associated with positive long-term outcomes for students, includ-
ing increased mathematics test scores (Attewell & Domina, 2008; Gamoran &
Hannigan, 2000); enrollment in advanced high school mathematics and science
course-taking sequences (Paul, 2005; Stein, Kaufman, Sherman, & Hillen, 2011);
and greater rates of college application, acceptance, and attendance (Atanda, 1999;
Spielhagan, 2006). Both access to the algebra course and the grade in which it is
taken are important factors in shaping a student’s academic trajectory. Taking alge-
bra in the eighth grade is crucial in order to allow students the time to complete a 4-
year, college preparatory sequence in high school (Loveless, 2008), which in turn
enhances their chances of college acceptance (Attewell & Domina, 2008), and sub-
sequent entrance into potentially high paying science, technology, engineering, and
mathematics (STEM) fields (Evan, Gray, & Olchefske, 2006).
The potential benefits of algebra enrollment in eighth grade, combined with
past racial and socioeconomic disparities in access to algebra courses (Adelman,
2006; Silva, Moses, Rivers, & Johnson, 1990; Stein et al., 2011) have fueled major
initiatives proposing early algebra-for-all students. The central idea is that provid-
ing the opportunity for all students to take algebra in eighth grade will ensure that
they are not excluded from college opportunities by course sequences that fall short
of college admission policy requirements (Liang, Heckman, & Abedi, 2012). There
is some evidence that nationally, placement into algebra is far less likely if a student
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 33
is Black or Hispanic, low SES, male, or from a single-parent household (Walston &
McCarroll, 2010). Although the “top-down” approach to mathematics equity by
requiring all students to acquire the same content has been criticized as lacking rel-
evance to the everyday lives of marginalized students (Martin, 2003), some policy
advocates have identified mathematics access as a civil rights issue, suggesting that
mathematics proficiency is a necessary component of literacy in the computer age
in order to fully participate in society (Moses, 1995; Moses & Cobb, 2001).
In this study, we compare students who performed poorly in eighth-grade al-
gebra to students who passed a lower-level, eighth-grade mathematics course, and
examine the academic and psychological and motivational factors associated with
success and failure. Such comparisons can be problematic due to a myriad of fac-
tors that may contribute to a given student’s success or failure in a course. For that
reason, in our analyses, we utilize propensity score matching to simulate random
assignment and to reduce possible selection bias on several demographic factors
that may influence student outcomes. Specifically, our analyses address the follow-
ing research questions:
1. How does the algebra proficiency level of students who took algebra in
the eighth grade and failed (defined in this study as receiving a grade of
“F” or “D”) compare to that of students who took a lower-level mathe-
matics course in eighth grade and passed (receiving a grade of “C” or
higher)?
2. How does the mathematics identification of students who failed algebra
in the eighth grade compare to the mathematics identification of those
who passed a lower-level, eighth-grade mathematics course?
3. How does the mathematics utility value (perceived usefulness) of stu-
dents who failed algebra in the eighth grade compare to the mathematics
utility value of those who passed a lower-level, eighth-grade mathemat-
ics course?
4. How does the mathematics interest of students who failed algebra in the
eighth grade compare to the mathematics interest of those who passed a
lower-level, eighth-grade mathematics course?
5. What is the relationship between eighth-grade course enrollment and
eleventh-grade “college readiness”?
Overview of the Literature
As a result of recent policy changes and initiatives, secondary students are
enrolling in advanced mathematics at levels never before seen in U.S. public edu-
cation. An examination of national longitudinal transcript data revealed an in-
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 34
crease in the number of mathematics courses taken from 2.7 total credits in 1982
to 3.6 credits in 2004 (Dalton, Ingels, Downing, & Bozick, 2007, p. iv). Over that
same period, there was a threefold increase in the number of students taking pre-
calculus and calculus, and a 50% drop in the number of students finishing high
school with Algebra I or less as their highest course taken (p. 12). Data from the
Early Childhood Longitudinal Study (ECLS-K) reveal that by the 2006–07 school
year, 39% of the study cohort was taking Algebra I or higher in the eighth grade
(Walston & McCarroll, 2010). In California, the percentage of eighth graders en-
rolled in algebra increased from 32% in 2003 to 59% by 2011, following the im-
plementation of policies designed to ensure that all eighth graders take algebra
(Liang et al., 2012). Moreover, there is some evidence that groups previously un-
derrepresented in higher mathematics courses have started to make inroads as a
result of the mandates in many urban districts requiring all students to be enrolled
in algebra in the eighth grade. A study of academic curricular intensity using tran-
script data from the National Educational Longitudinal Study (NELS:88) found
that Asians and Whites were taking the most demanding curricula overall. How-
ever, once SES and prior performance were controlled, results revealed that tradi-
tionally underperforming Black and Hispanic students had begun to take a more
demanding curriculum than White students (Attewell & Domina, 2008).
The shift towards increased student access to higher mathematics courses is
not without criticism. Significant debates have ensued among policymakers and
educators about the benefits and consequences of increasing early algebra readi-
ness and course-taking (Allensworth, Nomi, Montgomery, & Lee, 2009; National
Council of Teachers of Mathematics, 2008; Rosin, Barondess, & Leichty, 2009).
Allensworth and colleagues examined outcomes from a Chicago initiative man-
dating universal college preparatory coursework and framed the debate in terms
of a continuum, which has “constrained curriculum” advocates at one end and
“social efficiency” supporters on the other. Proponents of a social justice ap-
proach to curriculum (a position that Allensworth and colleagues term “the con-
strained curriculum”) contend that all students should experience a rigorous cur-
riculum that prepares them equally as well for both the workforce and higher edu-
cation. On the other end of the continuum, the social efficiency argument suggests
that, because students have different intellectual capacities, skills, and aspirations,
“schools have a duty to sort and match students to their future places in the social
and economic system” (p. 368). The social efficiency position seems to redefine
the role of public education as the “great sorter” rather than the “great equalizer.”
But from a social justice perspective, an important issue is whether this sorting is
based on academic merit as opposed to other factors.
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 35
Placement Factors and Algebra I Outcomes
Although more students are now being enrolled in higher-level mathematics
courses, some critics argue that all students might not be best served by taking Al-
gebra I early. Loveless (2008) maintains that the impetus for universal eighth-grade
algebra is an equity focus rather than one based on empirical evidence. However,
some empirical evidence suggests that certain groups have not received the same
access to algebra, even when their academic proficiency would seem to have pre-
dicted otherwise. An examination of course placement practices utilizing Early
Childhood Longitudinal Study (ECLS-K) data revealed significant disparities in
algebra course placements for Black students (Faulkner, Stiff, Marshall, Nietfeld, &
Crossland, 2014). High achieving Black students’ odds of being placed in algebra
were two-fifths lower than their White peers.
An earlier examination of the ECLS-K revealed that even though schools
were more likely to place their students with the strongest mathematics skills into
courses leading to algebra by the eighth grade, only 35% of Black students who
scored in the highest two nationally normed quintiles on a fifth-grade mathematics
test were placed into algebra in the eighth grade, compared to 63% of Whites, 68%
of Hispanics, and 94% of Asians (Walston & McCarroll, 2010). Higher achieving
male students moved on to eighth-grade algebra at lower rates than female students
(56% vs. 70%). In every quintile, students placed in algebra had higher subsequent
mathematics test scores than their counterparts placed in lower mathematics cours-
es, although the difference in the lowest quintile was not statistically significant. An
earlier study by Stone (1998) revealed that high SES students in a large urban dis-
trict who scored in the upper quartile on a standardized test of academic ability
were three times more likely to be enrolled in Algebra I than their low SES coun-
terparts scoring in the same quartile.
The disparity in algebra access for high achieving students raises a question as
to whether mathematics ability should be the sole determinant of access to college
preparatory coursework. Stereotypes and perceived abilities may play a role in
placement decisions that are not consistent with demonstrated performance. Well-
documented disparities in African American disciplinary actions, as well as their
overrepresentation in “judgment categories” of special education (T. C. Howard,
2010, p. 20), point to possible social and behavioral factors contributing to evalua-
tions of competence. The influence of such nonacademic factors in placement deci-
sions cannot be completely dismissed, especially for groups of students whose test
scores do not correlate with their opportunities. Evaluations of mathematics compe-
tence have also been found to be related to language proficiency levels, which can
be exacerbated by placement into tracked courses (Mosqueda, 2010). For English
language learners, bilingualism could be utilized as a resource as some have sug-
gested (Téllez, Moschkovich, & Civil, 2011), rather than a barrier to access.
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 36
In an examination of National Assessment of Educational Progress (NAEP)
data, Loveless (2008) highlighted the lack of correlation between enrollment in ad-
vanced courses and NAEP test scores, as well as a small decline in scores for stu-
dents in advanced courses over a period when more students were being granted
access to algebra. From among the students placed in advanced courses, he exam-
ined the characteristics of the bottom 10% of participating students in terms of per-
formance on the NAEP (extrapolated to represent approximately 120,000 students
nationwide) whom he refers to as the “misplaced” students. He found that a large
percentage of these students were poor minorities, with parents who did not gradu-
ate from college, and who had less qualified teachers while attending large urban
schools. He also noted that the large urban schools tended to shun tracking practices
in their course assignments, suggesting that such sentiments led to the misplace-
ment of these low-achieving students in the name of equity. Based on his analyses
of the students performing at the very bottom, he appeared to call for putting the
brakes on the algebra-for-all thrust:
Research exists showing that knowledge of algebra is essential for entry into occupa-
tions earning middle class wages. No evidence exists that it matters whether algebra is
learned in eighth grade or later, and some students may need more than a year to learn
the subject. (p. 12)
One can hardly argue with the assertion that students who are several years
behind grade level and enrolled in advanced courses are misplaced, especially if
they are given no support beyond that received by students performing at grade lev-
el. Policies that place students into courses for which they are underprepared cer-
tainly do not serve the ends of social justice, but rather potentially set up the stu-
dents for failure and make the work of teachers and other students more difficult.
However, a possible limitation in this argument is that the policy is being judged
based on the characteristics of a group of students that represented the lowest per-
forming 120,000 out of approximately 4.2 million eighth-grade students nation-
wide. By contrast, high achieving students who are denied access to Algebra I when
other similarly proficient students are granted access are arguably misplaced as well
(Walston & McCarroll, 2010). In both cases, it appears that the students are not be-
ing well served by their placements. Course placement decisions that are not con-
sistent with students’ demonstrated content knowledge give rise to policies de-
signed to ensure each student has access to courses of critical importance. However,
in the case of universal algebra access, the policy must be judged by its effects on
all student groups.
The research is mixed on whether low-performing students benefit from being
placed in higher mathematics courses. Data from the NELS:88 revealed that stu-
dents at all levels of performance benefitted from taking algebra in the eighth grade;
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 37
however, those in the lowest quintile benefitted the least (Gamoran & Hannigan,
2000). Achievement scores of ninth-grade Chicago students were unaffected fol-
lowing the enforcement of a policy mandate that required low-ability students to be
enrolled in Algebra I; however, over the long term, students in the lowest ability
groups were more likely to earn credits for upper-level mathematics courses than
they did prior to the implementation of the policy (Allensworth et al., 2009).
A review of 44 studies published from 1995 onward provides some indication
as to the causes for differential access; Stein and colleagues (2011) concluded: “Our
data suggest two major reasons for these demographic imbalances: underprepared-
ness and subjective placement factors” (p. 460). The fact that many students are not
prepared to take on algebra in eighth grade supports some critics’ argument that
simply placing students in higher-level mathematics courses without changing their
K–7 educational experiences is a recipe for failure (Liang et al., 2012; Schmidt,
2004; Williams, Haertel, Kirst, Rosin, & Perry, 2011). Nonetheless, when students
are among the highest achievers in their respective courses and still find themselves
denied access to this gatekeeper course, questions of equity in terms of course
placements naturally arise.
Failure Associated with Raising the Bar
Another major criticism of universal access mandates is the potential for in-
creased failure rates as a byproduct of moving students into algebra irrespective of
their proficiency levels. A mandate requiring all Chicago ninth-graders to take Al-
gebra I resulted in increased failure rates among students moved into algebra as a
result of the mandate (Allensworth et al., 2009). Dropout rates, however, did not
increase and mathematics scores on proficiency tests taken at the end of ninth grade
were unaffected. Increased algebra enrollment rates have been found to be associat-
ed with higher rates of failure in those, and subsequent mathematics courses, as
well as higher dropout rates later in high school (Silver, Saunders, & Zarate, 2008;
Waterman, 2010).
An examination of U.S. census data for individuals who graduated between
1980 and 1999 found that in states where mathematics and science course gradua-
tion requirements were mandated, dropout rates increased for all groups except
Hispanics, although Black and Hispanic male students were most severely impact-
ed. The dropout rate increase overall was less than a percentage point (0.82%), but
the increases for Black (1.88%) and Hispanic (2.58%) male students were the
greatest (Plunk, Tate, Bierut, & Grucza, 2014). These course graduation mandates
were also associated with a decrease in the likelihood for Black women and Hispan-
ic men to enroll in college, but no statistically significant impact to college enroll-
ment was observed on the overall sample.
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 38
Given that placing more students into a higher, more difficult mathematics
course results in more students failing the course, some have suggested placing
lower achieving students on a slower path (Liang et al., 2012; Loveless, 2008).
However, increased access to algebra has also resulted in more students scoring
proficiently on algebra knowledge measures, potentially leaving the gate open for
college access. For example, in California, the proportion of eighth-grade students
enrolled in Algebra I increased from 32% to 57% between 2003 and 2010, as did
the proportion of students scoring “proficient” or higher on the Algebra I California
Standardized Test (CST) (39% to 46%) over that same period (Williams et al.,
2011). Many students are taking advantage of the chance to succeed provided by
policy-mandated Algebra I access. The difficult decision arising from these circum-
stances is determining whether a policy designed to afford every student with an
opportunity for college access should be scrapped despite the advantages it provides
to many students. Conversely, a policy requiring all students to take algebra despite
data indicating that many are not prepared to succeed at it is difficult to defend from
a social justice perspective.
If universal access to algebra, regardless of prior achievement, is used as a
vehicle to ensure college preparation opportunities, some authors have contended
that it should include additional supports to scaffold learning for low achievers, as
well as support for teachers in learning how to handle heterogeneous student-ability
groupings. Nomi (2012) found the lack of these additional supports in the Chicago
algebra-for-all policy to be a “critical limitation” in its implementation (p. 501). She
asserted that such supports were instrumental in a somewhat successful “double-
dose” approach employed by Nomi and Allensworth (2009). In that approach, ninth
graders with lower test scores from the previous year were enrolled in both a regu-
lar algebra course and an algebra support course. Nomi suggested, as an alternative,
homogeneous groupings could be used while providing additional instruction time
for struggling student groups. In either grouping option, the additional support does
not necessarily need to be in the form of more time utilizing the same teaching
methods. Technology-based supports have shown promise in supporting underlying
basic mathematics skills in the context of a traditional mathematics classroom set-
ting (K. E. Howard, 2012) and can be utilized for those students who demonstrate a
need for them.
Addressing universal access with homogeneous groupings has been viewed as
problematic by critics who allege that lower achievers may be placed in courses that
are algebra in name only, providing a watered down curriculum that does not pre-
pare students as purported (Schneider, 2009). In that scenario, the same subjective
placement factors that have caused potentially algebra-ready students to be placed
in lower-level mathematics courses (Stein et al., 2011) may continue to limit oppor-
tunities to learn for capable students. In addition, given the negative academic con-
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 39
sequences and unequal access that have been associated with tracking practices in
the past (Mosqueda, 2010; Oakes, 2005), policies that separate students solely on
the basis of achievement levels have fallen out of favor. Conversely, mixed-ability
groupings present their own set of equity issues. Although such groupings may ad-
dress the needs of capable students who would have been denied access to algebra
under previous policies, inclusion of lower-achieving students, particularly those at
the lowest levels, may restrict the pace of education for higher achieving students.
Research is mixed as to the impact that heterogeneous ability groupings have on the
most advanced students. Some researchers’ findings have suggested that the per-
formance of higher achieving students is compromised by de-tracking lower per-
forming students into more advanced courses (Allensworth et al., 2009; Loveless,
2009; Nomi, 2012), presumably due to the need for teachers to slow the instruction-
al pace or otherwise adjust content delivery for a wider range of proficiency. Other
researchers suggest that not only do lower-achieving students benefit from de-
tracked courses, but also higher-achieving students have been found to achieve at
even higher levels in mixed-ability groups than in high-level tracked groups (Boal-
er, 2011). However, as discussed next, there is some evidence that not all students
in higher achieving homogeneous classrooms benefit from this arrangement.
Comparing data across two Chicago initiatives, Nomi and Allensworth (2014)
found that although test scores were higher on average when course enrollments
were sorted by student skill levels, grades and pass rates in high-skill courses actu-
ally declined. Nomi and Allensworth suggested that these declines were due to
grading practices (e.g., on a curve), higher demands, and high-achieving students
finding themselves below average compared to their classroom peers. Consequent-
ly, students with skills near the bottom of their respective courses had higher failure
rates. On the other hand, low-skilled students’ mathematics scores declined slightly
after sorting when no changes were made to curriculum, but increased considerably,
along with grades and pass rates, when more classroom time and teacher profes-
sional development resources were provided. In addition, low-skilled students were
more likely to experience disruptive classroom environments due to more behavior-
al problems when sorted by skill level.
Whether students are sorted by skill levels or placed in mixed-ability group-
ings, raising the bar on the difficulty level of eighth-grade courses will likely result
in increased failure rates in the short term, especially when the structure of the pre-
ceding K–7 instruction has not yet been adequately redesigned to better prepare
students for the increased demands (Schmidt, 2004). If students, for whatever rea-
son, find themselves unprepared to pass algebra in the eighth grade, would it be bet-
ter to place those students in a lower-level mathematics course instead? Are there
academic benefits to at least exposing eighth graders to algebra regardless of their
preparedness, even if they fail the course? In other words, can we find a measure of
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 40
success associated with course failure? What about the psychological and motiva-
tional implications of such an approach? Research indicating the relationship be-
tween higher-level course-taking and higher levels of mathematics achievement has
not often examined the impact of failure in comparison to successfully completing a
lower-level course, nor has it specifically focused on the psychological outcomes
associated with success and failure.
A comparison of proficiency rates among Californian, ninth-grade students
who failed (scored less than proficient) the previous year’s Algebra I CST to those
who passed the CST for General Mathematics in eighth grade (scored proficient or
above) revealed stark differences in the two groups: the students who had failed the
Algebra I CST had 69% less of a chance of passing it in the ninth grade than those
who had passed the General Mathematics CST in the eighth grade (Liang et al.,
2012). At first glance, this would seem to support placing more students in General
Mathematics courses; however, upon closer examination only 27.5% of the students
in General Mathematics passed the CST in the eighth grade (as opposed to a 41.8%
pass rate for Algebra I). The group of students who passed the General Mathemat-
ics CST likely included some misplaced students, who may have scored well
enough to be placed in algebra but were sorted out by subjective factors previously
discussed. Nonetheless, the high rate of repeated failure (5 out of 6) for those who
had failed the Algebra I CST the previous year cannot be ignored. This rate indi-
cates that the vast majority of students repeating algebra did not benefit enough
from the previous year’s exposure to score proficiently the second time around. An
alternative explanation is that many were not motivated enough to fully apply
themselves either time they took the course, which may be the case for students
who have chronically failed mathematics during their short K–8 experience but
continued to advance due to social promotion practices (Zinth, 2005).
The Psychology of Failure
Examining the impacts of failing algebra or passing a lower-level course, all
other factors being equal, is an essential element to assessing the wisdom of an al-
gebra-for-all policy. The direct academic impacts of such a policy may include
lower test scores and the narrowing of the window available for completion of a
college preparatory course sequence. Possible indirect psychological impacts are
less obvious, but can be just as detrimental to students’ future academic success.
These psychological effects may include identifying less with the mathematics do-
main, having a lower perceived value for mathematics, and experiencing a decline
in overall interest in mathematics (Crocker, Major, Schmader, Spencer, & Wolfe,
1998; Keller, 2007; Spencer, Steele, & Quinn, 1999).
Several psychological theories have addressed identification with mathemat-
ics among underrepresented groups by illuminating coping responses to stigmatiza-
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 41
tion and societal stereotypes (Major & O’Brien, 2005; Steele & Aronson, 1995).
Socially stigmatized groups have exhibited patterns of underperformance when
group membership was made salient in testing environments—a phenomenon re-
ferred to as “stereotype threat” (Steele, 1997). The effects of stigma, or an attribute
or identity that is devalued or undesirable in a social context, can impact the psy-
chological, educational, and social outcomes of stigmatized group members
through confirming negative stereotypes associated with group membership (Steele,
2003; Steele & Aronson, 1995), disengagement or “disidentification” (Crocker et
al., 1998), and distancing one’s identity from the target domain (Crocker, Major, &
Steele, 1998). If disidentification with the mathematics domain were to occur fol-
lowing failure in eighth-grade algebra, it would likely influence the students’ psy-
chological dispositions towards mathematics courses in subsequent years, offering
some explanation for the observed repeated failure rates.
Identifying with a domain is not, in and of itself, sufficient to motivate stu-
dents to engage in the work necessary to excel in that domain. Wigfield and Eccles
examined children’s mathematical beliefs through their expectancy-value theory
(Wigfield, 1994; Wigfield & Eccles, 2000), and posited that expectancy and task
value are the two most influential predictors of achievement behavior. The expec-
tancy component refers to a student’s beliefs as to whether they are able to perform
a task (expectation level for success), whereas the value component addresses their
beliefs about why they should perform the task at all (Pintrich & Schunk, 2002;
Schunk, Pintrich, & Meece, 2008). The influence of negative stereotypes on student
expectancy has been demonstrated as early as middle school with Black, Latina/o,
and low SES students (K. E. Howard & Anderson, 2010), the very groups that have
been found to be most frequently represented among the lowest performing stu-
dents in advanced courses (Loveless, 2008).
The value students ascribe to the mathematics domain is an important marker
of student engagement insofar as it is a strong predictor of whether students decide
to apply themselves to their mathematics courses. In the expectancy-value model,
the value component has two subcomponents (among others) that may be affected
by failing a course: intrinsic value and utility value (Schunk et al., 2008). Intrinsic
value is the student’s subjective interest or enjoyment in doing a task, which is in-
fluenced by past performance in that task or domain (Eccles & Wigfield, 2002).
Utility value is the perceived usefulness of a task in helping the student to meet his
or her future goals; these goals can be adjusted downward as a result of self-
regulatory processes following failure (Schmitz & Perels, 2011). These two sub-
components of task value are of particular interest in this study as they may help to
illuminate possible indirect effects of universal access policies that may not be evi-
dent in immediate academic outcomes. Understanding how these variables are im-
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 42
pacted by placement decisions, as well as by subsequent success or failure, can clar-
ify the wisdom and consequences of enacting particular course-taking policies.
It has been said that a society can be judged by how it treats its most vulnera-
ble members. Policymakers must weigh the impacts of a mandate that requires all
students to follow a particular curriculum against those of a policy that does not,
particularly on students whose futures are impacted most severely as a result. Thus,
research is needed to examine the impact of failure on students’ longitudinal math-
ematics achievement and attitudinal outcomes, and to determine if higher-level
course placements benefit students from all levels of attainment. This study is an
examination of the impact of eighth-grade mathematics course enrollment and suc-
cess and failure on subsequent algebra proficiency, college readiness, and psycho-
logical and motivational dispositions.
Methods and Data
A nationally representative sample was used for this study (Ingles et al., 2014;
National Center for Education Statistics, 2011) to examine the relationships among
mathematics course success, failure, and achievement outcomes, as well as to ex-
amine the levels of domain identification, utility, and interest associated with stu-
dents who fail eighth-grade Algebra I. The High School Longitudinal Study of
2009 (HSLS:09) was conducted by the U.S. Department of Education National
Center for Educational Statistics. We examined base-year student data, which were
collected from ninth graders during the fall of the 2009–10 school year, and the first
follow-up wave of data collected in 2012 when most of the students were in the
spring semester of eleventh grade. The HSLS:09 data were derived from a sample
consisting of more than 21,000 students from 944 public, charter, and private
schools in the United States. Each wave of data collection included a mathematics
assessment of algebraic reasoning and a computer-based survey addressing various
psychological and motivational constructs (Ingels, Dalton, Holder, Lauff, & Burns,
2011; Ingels et al., 2014).
Data Sources
All of the analyses in this article were based on data obtained from the stu-
dent-level public use files for the HSLS:09 (base year) and the HSLS:12 (first fol-
low up) data sets. The HSLS:09 variable identifying the highest course each partic-
ipant took in the eighth grade (S1M8) includes nine self-reported options, including
courses more advanced than Algebra I. We were only interested in comparing stu-
dents who took a version of Algebra I with those who took a lower mathematics
course; therefore, we initially filtered the 21,444 cases in the complete data set to
include only the students who reported taking Algebra I (including IA and IB; n =
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 43
6,675) in Group 1, and those who reported taking Math 8, Advanced/Honors Math
8 (not including Algebra), or Pre-algebra (n = 12,254) in Group 2. The HSLS:09
variable identifying the self-reported grade each participant received in their highest
level mathematics course taken in the eighth grade (S1M1GRADE) was used to
filter Group 1 to only include the students who received a grade of “D” or “F” in
Algebra I, which resulted in 337 students (5.1%). Group 2 was filtered to include
only the students who received a grade of “C” or higher in a course lower than Al-
gebra I (i.e., Math 8, Advanced or Honors Math 8, or Pre-algebra), which totaled
11,000 (89.8%). Finally, both groups were trimmed to exclude any cases which had
missing data for any of the variables being considered, which resulted in a Group 1
total of 274 and a Group 2 total of 8,504.
Propensity Score Matching Procedures
Given that our data were compiled from non-randomized cases, propensity
score matching was utilized to achieve balance on several observed covariates
(Stuart, 2010) and to reduce the impact of treatment-selection bias in the estimation
of treatment effects. Our final data set (before matching) contained 274 cases of stu-
dents who received a D and/or F in their respective Algebra I course, and 8,504 cases
of students who received an A, B, or C in their lower level mathematics course in
eighth grade. The covariates selected for generating the propensity scores were the
HSLS:09 student-level data file variables for students’ sex (X1SEX), race
(X1RACE), language status (X1DUALLANG), locale (X1LOCALE), region
(X1REGION), socioeconomic status quintile (X1SESQ5), along with a dummy vari-
able for participant’s test date. The language status variable identifies the participant’s
first language as either English, non-English, or equally English and a non-English
language. The locale variable identifies the participant’s address as city, suburb,
town, or rural. The region variable identifies the participant’s location in the country
as northeast, midwest, south, or west. The SES quintile variable is a composite varia-
ble derived from the parent/guardians’ education, occupation, and family income.
The test date variable identifies the date when the study-administered test of algebraic
reasoning was completed. The tests were administered on three dates during the
2009–10 school year, providing participants with different levels of exposure to their
ninth-grade curriculum prior to taking the test. This variable was included in the
matching procedure to control for the amount of time elapsed between the date when
the participants received their letter grade in the eighth grade and the date when they
took the test in the ninth grade.
The propensity score matching procedure was conducted in SPSS 21, using an
“R” plugin, which allows estimation of propensity scores using logistic regression (R
Development Core Team, 2008). The propensity scores represent the probability that
a person or case will land in the treatment condition given the values on all of the co-
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 44
variates. In this study, the propensity scores represent the probability that a student
would fail Algebra I, given the values on the seven included covariates. Once the
propensity scores are computed, the program utilizes three “R” packages to perform
the procedure: “MatchIt” (Ho, Imai, King, & Stuart, 2007), “RItools” (Bowers,
Fredrickson, & Hansen, 2010), and “cem” (Iacus, King, & Porro, 2009). The match-
ing technique utilized in this study was 1:1 nearest neighbor matching (without re-
placement), meaning each “treated” participant was matched to a single “untreated”
participant who had the propensity score that was closest to an exact match. We used
a caliper of .2 of the standard deviation of the logit of the propensity score to ensure
good matches (Thoemmes & Kim, 2011). The caliper defines the maximum allowa-
ble difference in estimated propensity scores for matches. Cases outside of the com-
mon support area (region of distribution in which cases from both groups are ob-
served) were discarded for the larger (untreated) group only, in order to maintain a
sufficient sample size for subsequent statistical analyses.
After propensity score matching, the treatment and control groups each consist-
ed of n = 274 students. An overall imbalance measure (Hansen & Bowers, 2008) was
not significant, X2(7) = 1.35, p = .99, indicating good balance after matching. The his-
tograms of the standardized differences depicted in Figure 1 illustrate greatly im-
proved covariate balance after matching as compared to the unmatched sample. The
matched sample histogram is heavily centered on zero, indicating no systematic dif-
ferences existed after matching. Figure 2 depicts the standardized mean differences
for each of the covariates before and after matching, illustrating that the magnitude of
the standardized differences overall were greatly improved after matching. The
standardized mean differences of all covariates were close to 0 after matching.
Figure 1. Histograms with overlaid density estimates of standardized differences
before and after matching.
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 45
Figure 2. Dotplot of standardized mean differences (Cohen’s d) for all covariates
before and after matching.
Measures
Algebra Proficiency
An online mathematics test of algebraic reasoning was completed by each of
the HSLS:09 participants during the fall semester of the ninth grade and during the
spring semester of the eleventh grade. The instrument used to measure algebra profi-
ciency was developed and validated by the American Institutes of Research (Ingels et
al., 2011). The test and item specifications addressed six algebraic content domains
(the language of algebra; proportional relationships and change; linear equations, ine-
qualities, and functions; nonlinear equations, inequalities, and functions; systems of
equations; and sequences and recursive relationships) and four algebraic processes
(demonstrating algebraic skills; using representations of algebraic ideas; performing
algebraic reasoning; and solving algebraic problems). Item analyses were conducted
to compile a pool of optimally performing items, including differential item function-
ing (DIF) statistics to detect potential racial/ethnic and gender biases.
Base year. The base year assessment administered to the participants contained
40 items and employed an adaptive design. There were 72 unique items used to com-
pile all of the variations of the test. The item response theory (IRT) estimated reliabil-
ity of the base year instrument was 0.92 after applying sample weights (Ingels et al.,
2011). The HSLS:09 base year variable X1TXMSCR represents an IRT-based esti-
mate of the score for each participant on the full set of 72 items. Each participant
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 46
completed the assessment on one of three different test dates during the fall term of
his or her ninth-grade school year (September 2009, October 2009, or January
2010). The HSLS:09 base year theta score variable X1TXMTH is an ability esti-
mate score based on the same metric as the IRT item-level difficulty parameters. It
provides a summary measure appropriate for longitudinal analyses to measure
growth in achievement over time.
First follow-up. A second algebra assessment addressing the previously listed
six algebraic content domains and four algebraic processes was administered during
the spring semester of the study cohort’s junior year. The follow-up theta score var-
iable (X2TXMTH) was used to compare participant scores over both waves as the
common items between waves allowed for equating the IRT scores across waves.
The follow-up assessment contained a 69-item pool, including 23 common items
across the two waves. This item pool also included 20 new items that were field
tested and added to include higher difficulty items and guard against a ceiling ef-
fect. The IRT-estimated reliability of this follow up assessment was 0.92 after sam-
ple weights were applied (Ingels et al., 2014).
Psychological and Motivational Scales
Several scales were created for the HSLS:09 and included in the field-testing
for the algebra proficiency variables (Ingels, Herget, Pratt, Dever, & Copello,
2010). Three of those scales were examined as part of our analyses over both waves
of data. All of the data for the scales were reported in the HSLS:09 in standardized
form (with a mean of 0 and a standard deviation of 1) to allow for combining varia-
bles with different scales during analyses. A threshold of .65 was used for inclusion
of each of the scales as an adequate measure of internal consistency and each of the
scales performed above this threshold in both waves of data collection. Table 1
provides coefficients of reliability for each of the scales examined over each wave
of data (Ingles et al., 2011; Ingels et al., 2014).
Mathematics identity. The HSLS:09 mathematics identity scale score was
used to measure students’ mathematics identity during both waves of data collec-
tion (X1MTHID/X2MTHID; Cronbach’s Alpha .84 and .88, respectively). This
scale measures the participant’s level of agreement from 1 (strongly agree) to 4
(strongly disagree) on two statements: “You see yourself as a math person” and
“Others see me as a math person,” with higher values indicative of higher mathe-
matics identity.
Mathematics utility. The HSLS:09 mathematics utility scale score was used in
both waves of data collection to measure participants’ beliefs in the utility value of
learning mathematics (X1MTHUTI/ X2MTHUTI; Cronbach’s Alpha .78 and .82,
respectively). This scale measures the participant’s level of agreement from 1
(strongly agree) to 4 (strongly disagree) on three statements about the importance of
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 47
mathematics for their future: “My math class will be useful for college”; “My math
class will be useful for everyday life”; and “My math class will be useful for a fu-
ture career.” Higher values indicate higher mathematics utility.
Mathematics interest. The HSLS:09 mathematics interest scale score was
used for both waves of data collection to measure participants’ interest in their cur-
rent mathematics course (X1MTHINT/X2MTHINT; Cronbach’s Alpha .75 and .69,
respectively). This scale includes two questions inquiring as to the students’ favor-
ite and least favorite (wave 1 only) school subjects, followed by four statements
about their current grade mathematics course assessing their level of agreement
from 1 (strongly agree) to 4 (strongly disagree): “You are enjoying this class very
much”; “You think this class is a waste of your time”; “You think this class is bor-
ing”; and “You really enjoy math.” Higher values indicate higher mathematics in-
terest. The first year follow-up instrument for mathematics interest was revised to
eliminate a question asking for the student’s least favorite subject, but still showed
adequate internal consistency.
College Readiness Variables
Three variables measuring college readiness were examined by group and
compared to the entire HSLS:12 sample. Each variable corresponds to a single
question that was part of the questionnaire in the follow-up data collection in the
fall semester of the participants’ eleventh-grade school year.
Community college. The HSLS:12 variable S2REQ2YR corresponds to the
question “By the summer of 2013, do you think you will have met the minimum
requirements needed for admission to a 2-year community college?” The response
options were: Yes, No, and Don’t Know.
Typical four-year college. The HSLS:12 variable S2REQTY4YR corresponds
to the question “By the summer of 2013, do you think you will have met the mini-
mum requirements needed for admission to a typical 4-year college?” The response
options were: Yes, No, and Don’t Know.
Selective college. The HSLS:12 variable S2REQSEL4YR corresponds to the
question “By the summer of 2013, do you think you will have met the minimum
requirements needed for admission to a highly selective 4-year college such as Har-
vard University?” The response options were: Yes, No, and Don’t Know.
Results
Descriptives
Statistical analyses were performed on the final base-year set of 548 partici-
pants, split evenly between students who took and failed algebra in eighth grade
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 48
and those who took and passed a lower-level mathematics course in eighth grade.
As illustrated in Table 1, the groups were well balanced through propensity score
matching on gender, race, language status, and socioeconomic status. The balance
on these important variables was maintained for the participants in the matched
group who provided responses in the follow-up data collection.
There was a response rate of 83.8% (n = 459) for the follow-up wave mathe-
matics assessment for this group of participants, which compared favorably to the
overall unweighted follow-up response rate of 84.3%. Little’s MCAR statistic
(SPSS Missing Values 22.0) revealed that the missing data met the assumption of
MCAR, 2(39) = 52.84, p = .07. There were no systematic patterns of missing data
when compared to the observed values for all of the matched covariates, the prior
mathematics assessment and psychological measure scores, and the college-bound
variables.
Table 1
Post-Matching Demographic Characteristics of the Treatment (Algebra Fail)
and Control (Lower-Level Math Pass) Groups
Base Year
Mathematics Assessment
Follow Up
Treatment
n = 274
Control
n = 274
Treatment
n = 228
Control
n = 231
Gender
Male 62% 62% 62% 63%
Female 38% 38% 38% 37%
Race/Ethnicity
White 47% 49% 45% 50%
Black 10% 10% 9% 10%
Latino 26% 20% 26% 19%
Asian 3% 8% 3% 8%
Other 14% 13% 17% 13%
First Language
English Only 80% 78% 79% 78%
Non-English 13% 14% 14% 12%
English and
Non-English 7% 9% 7% 10%
SES quintile
First 24% 25% 25% 23%
Second 22% 19% 21% 18%
Third 20% 24% 21% 26%
Fourth 19% 18% 19% 17%
Fifth 15% 15% 14% 17%
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 49
Algebra Proficiency
To examine for algebra proficiency differences between to two eighth-grade
course conditions (failing algebra or passing lower-level course) and any longitudi-
nal differences, a mixed between-within subjects analysis of variance was conduct-
ed. The participants’ theta scores on the test of algebraic reasoning were compared
across the two waves of data. There was no significant interaction between condi-
tion and time: Pillai’s Trace = .002, F(1, 457) = .80, p = .37, partial η² = .002. As
expected, there was a significant main effect for time: Pillai’s Trace = .32, F(1, 457)
= 218.50, p < .001, partial η² = .32. Both groups showed statistically significant in-
creases in scores over time. The main effect comparing the two course conditions
was not statistically significant: F(1, 457) = .92, p = .34, partial η² = .002. Whereas
both groups significantly increased their mean theta scores from the base year to the
follow-up data collection, there were no statistically significant differences in
scores between the two groups.
Psychological and Motivational Outcomes
The study participants were measured on psychological and motivational
scale variables of interest at each of the two waves of data collection. Repeated-
measures MANOVA analyses revealed a significant multivariate effect for course-
taking group: Pillai’s Trace = .03, F(3, 362) = 3.86, p = .01, η² = .03; and a signifi-
cant interaction effect between course assignment and time: Pillai’s Trace = .03,
F(3, 362) = 3.37, p = .02, η² = .03. Univariate tests revealed significant interaction
effects between course assignment and time for mathematics identity F(1, 364) =
7.05, p < .01, η² = .02; mathematics interest F(1, 364) = 4.86, p = .03, η² = .01; and
mathematics utility F(1, 364) = 4.11, p = .04, η² = .01. These were small effect siz-
es.
To further examine the interaction effects, the data set was split and MANO-
VAs were performed for each wave of data (Time 1 in ninth grade; Time 2 in elev-
enth grade) separately. There was a statistically significant difference between the
groups on the combined dependent variables for the base year data: F (3, 544) =
14.38, p < .001; Pillai’s Trace = .07; partial η² = .07. The eta squared of .07 indi-
cates a medium effect size (Cohen, 1988). The difference on each of the three de-
pendent variables reached statistical significance when the data were separately
analyzed. The students who took Algebra I and failed reported lower ratings in the
ninth grade on mathematics identity F(1, 546) = 23.44, p < .001, partial η² = .04;
mathematics interest F(1, 546) = 24.17, p < .001, partial η² = .04; and mathematics
utility F(1, 546) = 31.88, p < .001, partial η² = .06 than students who had passed a
lower-level, eighth-grade mathematics course. The mean scores for the differences
by group are displayed in Table 2, along with the weighted mean scores for students
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 50
who passed Algebra I in the eighth grade. The effect sizes for mathematics identity
and mathematics interest were small, whereas the effect size for mathematics utility
was moderate (Cohen, 1988).
Table 2
Means and Standard Deviations for Psychological and Motivational
Dependent Variables for Treatment Group (Algebra Fail), Control Group
(Lower Level Math Pass), and Algebra Pass Group
Treatment
(failed algebra)
n = 274
Control
(passed lower math)
n = 274
Passed Algebraa
n = 6240bc
Dependent
Variable
M SD
M SD
M SD
Mathematics
Identity -.44 .99 -.03 .98 .32 .93
Mathematics
Interest -.33 1.05 .11 1.05 .14 .98
Mathematics
Utility -.26 1.11 .23 .92 -.09 .97
Note. Scale values were standardized to a mean of 0 and standard deviation of 1.
aNo statistical comparison analyses were performed using the “Passed Algebra” group due to unequal sample
sizes; statistics for this group are provided only for reference purposes.
bCases weighted by base year student weight (w1student) to adjust for oversampling of specific subgroups.
cThis group was matched on covariates, therefore no filtering of the data occurred for these students.
Data from the second wave were examined on the same variables. The miss-
ing/non-response levels for mathematics identity, interest, and utility in the follow-
up wave of data were 18.1%, 32.3%, and 18.8%, respectively. The non-response
level for interest was higher due to the fact that interest in current mathematics
course did not apply to all respondents (i.e., this item was skipped for any student
who was not enrolled in any mathematics course at the time). Although the missing
data were found to be non-systematic in the MCAR analysis previously noted, re-
sponse rates under 80% fail to meet researchers’ recommended acceptance levels of
ignorable missing data (Sterner, 2011; Vriens & Melton, 2002). For such instances,
imputation methods represent a more favorable approach than traditional methods
such as deletion (casewise or pairwise) or mean substitution (Acock, 2005). Imputa-
tion allows the researcher to analyze a complete data set and prevents the loss of
statistical power that results from the elimination of data. The missing values in this
data set were replaced using multiple imputation. The analyses were performed us-
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 51
ing the original data set, but were repeated using the imputed data sets for confirma-
tion purposes.
One-way between-groups multivariate analyses of variance (MANOVAs)
were performed on both the original data (with casewise deletion) and on the full
imputed data sets to examine the group differences on the psychological and moti-
vational scale variables of interest for the second wave of data collection. For the
original (non-imputed) data set, there was no statistically significant difference be-
tween the groups on the combined dependent variables: F(3, 362) = .84, p = .47;
Pillai’s Trace = .01; partial η² = .007. Summary mean statistics of the five complete
data sets generated using multiple imputation also revealed no statistically signifi-
cant differences in the two groups: F(3, 546) = 1.74, p = .20; Pillai’s Trace = .02;
partial η² = .009. Figures 3, 4, and 5 visually depict the changes in students’ math-
ematics identity, interest, and utility from the base year data collection in ninth
grade to the first follow-up wave in eleventh grade using the original data set.
Whereas students who failed Algebra I in eighth grade exhibited lower scores on
psychological and motivational scale variables than those who passed a lower
course in wave 1 (ninth grade), these differences were no longer present by the sec-
ond wave (eleventh grade).
Figure 3. Line plot of change in mathematics identity
from base year to first follow up.
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 52
Figure 4. Line plot of change in mathematics interest
from base year to first follow up.
Figure 5. Line plot of change in mathematics utility
from base year to first follow up.
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 53
College Readiness Data Analyses
Data from the follow-up survey were examined to determine whether there
were differences between course enrollment groups in their expected college quali-
fication rates. The nonresponse rate for the three college readiness variables ranged
from 18.2% to 19.3%. Chi-square tests of independence were performed to examine
the relation between eighth-grade course enrollment and reported on-track status for
meeting college requirements in the second semester of eleventh grade. The relation
between course enrollment and meeting requirements for a 2-year college was not
statistically significant: 2(1, N = 548) = .01, p = .93, phi = -.01. The relation be-
tween course enrollment and meeting requirements for a typical 4-year college was
statistically significant: 2(1, N = 548) = 7.52, p < .01, phi = -.12. Students who
failed eighth-grade algebra were less likely to report being on track for meeting the
requirements to attend a typical 4-year college by the end of their senior year than
students who passed a lower-level, eighth-grade course. The relation between
course enrollment and meeting requirements for a selective 4-year college was not
statistically significant: 2(1, N = 548) = 1.2, p = .27, phi = .06, although relatively
few students from either group are represented in this category. Table 3 displays the
percentages (by college type) of each group that expected to meet college require-
ments by the end of their senior year. The same statistics are provided for students
who passed Algebra I in the eighth grade, as well as for the entire HSLS:09 sample.
Table 3
Percentage of Students Who Will Meet College Requirements
Group 2-Year college Typical 4-Year Selective 4-Year
Failed Algebra I
70.9
44.8
9.1
Passed Lower
Mathematics Course
77.7 57.3 5.1
Passed Algebra I 90.2 80.7 16.8
Entire HSLS:09
Samplea
79.8 65.4 13.0
Note. Cases weighted by first follow-up student weight (w2student) to adjust for oversampling of
specific subgroups.
aThis group was matched on covariates; therefore, no filtering of the data occurred for these stu-
dents.
Discussion and Implications
The HSLS:09 data show that students who failed Algebra I in eighth grade
did not score significantly differently in ninth grade on a test of algebraic reasoning
from students who passed a lower-level mathematics course in the eighth grade,
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 54
when matched on several covariates. When compared to the matched sample, the
students who failed Algebra I in the eighth grade did not demonstrate any immedi-
ate proficiency benefits from their exposure to Algebra I content. This finding is
consistent with previous studies’ findings that enrollment in college preparatory
mathematics courses did not impact academic test outcomes (e.g., Allensworth et
al., 2009; Loveless, 2009). Some prior research suggests that residual benefits from
exposure to algebra content may not become evident until after subsequent algebra
course-taking has occurred. For example, Gamoran and Hannigan (2000) observed
achievement growth between grades 8 and 10 from exposure to algebra, and At-
tewell and Domina (2008) found an advanced high school curriculum to be associ-
ated with higher test scores in the twelfth grade. However, the participants in this
study who had been enrolled in eighth-grade algebra did not demonstrate any resid-
ual performance benefits by the eleventh grade, as they did not perform statistically
differently from those who had been enrolled in a lower eighth-grade mathematics
course. It should be noted that, without having access to performance data for our
sample prior to the eighth grade, we could not state definitively that the groups did
not differ in proficiency level prior to receiving their failing or passing grades.
Nonetheless, their measured proficiency rates did not differ in the ninth and elev-
enth grades.
Although the students’ exposure to eighth-grade Algebra I did not affect test
scores, failing the course was associated with lower levels of mathematics identity,
interest, and utility in the following year as compared to students who passed the
lower-level course. However, by the second half of eleventh grade when the second
wave of data was collected, these two groups were no longer statistically different
on these psychological measures. Although it might be encouraging to attribute this
outcome to resiliency on the part of the failing students, an examination of the uni-
variate outcomes suggest that this inference is only partly accurate. Whereas the
mathematics identity scores for the failing students rebounded to be statistically at
the same level as those who had passed a lower course, the attenuation of the gap in
scores for interest and utility was more the result of declines on the part of the stu-
dents who had passed the lower level course.
It is important to note that it cannot be inferred that the more negative psycho-
logical scores were necessarily caused by the failing grades. It is feasible that the
students failed because they already had more negative dispositions toward mathe-
matics, or conversely that negative dispositions emerged following the failure. Re-
gardless of the direction of causality (if any), the existence of more negative dispo-
sitions towards mathematics indicates that psychological dispositions are somehow
associated with mathematics performance in the eighth grade, and placing these
students in advanced mathematics courses without addressing these dispositions
may be inadequate in terms of the support they may need. The results also provide
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 55
an argument against placing students in the same course the year following a failure
without attempting to address the motivational characteristics that may have con-
tributed to their failure.
One interpretation for these findings is that the misplacement of students in
eighth-grade Algebra I may have detrimental psychological effects on underpre-
pared students (Loveless, 2008). This interpretation has implications for policy and
practice, especially considering the fact that there does not appear to be any aca-
demic advantage gained by such placements after enduring the adverse psychologi-
cal effects. Rationale for placing students in courses in which they are (might be)
unable to pass would need to be weighed against the possible impact of failure on
future interest in mathematics, and ultimately on mathematics performance. To that
point, the higher mathematics utility and interest ratings for the students enrolled in
lower mathematics courses did not persist, but rather declined to a level not statisti-
cally different from the students who failed Algebra I by the time they reached
eleventh grade. A possible silver lining is that with respect to mathematics identity,
any impact from failure does not appear to be permanent as the failing Algebra I
students recovered by the time the second wave occurred.
On the critical issue of college readiness the results were somewhat mixed.
The groups did not differ statistically in terms of being on track to meet the re-
quirements of a 2-year college, which generally speaking, usually equates to having
a high school diploma or equivalent (College Board, 2012; Learn.org, 2015). Over
70% of students from each group reported meeting the requirements to attend a 2-
year college. In contrast, students who failed Algebra I were significantly less likely
than those who passed a lower mathematics course to be on track for meeting the
requirements of a typical 4-year college by the end of their senior year. The stu-
dents in this study were the highest performing among those placed in lower math-
ematics courses and the lowest performing of those placed in Algebra I. These re-
sults are consistent with Nomi and Allensworth’s (2014) findings of better college
readiness for students just below the cut-score for sorting into different courses, as
opposed to those just above it.
The results of this study are also consistent with the gatekeeper notion that ex-
iting eighth grade without passing Algebra, whether due to a failing grade or non-
enrollment in the course, negatively impacts one’s chances for meeting the re-
quirements for admission into college (Atanda, 1999; Spielhagan, 2006). The high-
er college readiness rates for students who passed a lower-level course would seem
to provide some support for putting underprepared eighth-graders on a slower path
(Liang et al., 2012; Loveless, 2008), except for the fact that in both cases they fell
far short in college readiness of both the national average and the students who
passed Algebra I. Whereas passing the lower course provides better odds than fail-
ing Algebra I, it seems to represent a band-aid solution for a much larger issue—
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 56
inadequate preparation for the required intellectual demands of algebra during these
students’ K–7 experiences. It apparently does matter whether algebra is learned in
the eighth grade or later. Moreover, to engage in practices or enact policies that de-
lay the acquisition of content that predicts college readiness is to, in effect, build
inequality into the educational structure. In urban schools with large populations of
African American, Latina/o, and lower SES students, perceptions about ability may
lead well-intentioned policymakers to give students the “gift of time” by delaying
their entry into algebra, but the end result of such a gift may be a life trajectory ab-
sent of higher education and all the benefits that accompany it.
The results on the psychological and motivational scales do not appear to
support the idea of placing some students on a slower track as a long-term solution
either. The differences between the groups on these measures all but dissipated by
the time they reached eleventh grade, and neither group approached the college
readiness rate that students passing Algebra I attained. Given that eighth-grade Al-
gebra I success equates to significantly better odds of college readiness, an ap-
proach systematically denying some students access by building slower paths into
the educational structure is not supported by our findings, especially when others
have found increased access to result in more students passing algebra (Williams et
al., 2011).
The focus in this study has been placed on the Algebra I students who failed,
but we must keep in mind that 95% of the students taking Algebra I reported pass-
ing the course, putting them among the group that reported a 4-year college readi-
ness rate of 80%. A limitation that must be acknowledged in this study is the fact
that the propensity score matching technique utilized only accounts for the selection
bias on variables observed in the HSLS data set, but not for any unobserved varia-
bles that may have influenced the outcomes analyzed. In addition, our focus on
those students who self-reported failing their courses severely limited our sample
size to a fraction of the overall study sample. As a result, the impact of failing alge-
bra observed for this subsample on subsequent measures cannot be generalized to
the vast majority of students who were not analyzed in this study. It is plausible,
and even likely, that psychological measures for students who passed their courses
would be impacted differently had these students experienced failure in the course,
especially if course failure is not something they have encountered previously.
These limitations notwithstanding, in order for policymakers to respond to critics
that point to increased failure rates and the resulting psychological effects as justifi-
cation for eliminating universal algebra access policies, understanding the possible
impacts of these failures is critical. In addition, understanding the effect that placing
struggling students in lower mathematics courses can have on college-readiness in-
forms decisions on whether to support universal access for all students.
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 57
The algebra-for-all policies that have been implemented in recent years ap-
pear to have arisen from the hope that providing access to this important course will
move our educational system a little closer to Horace Mann’s notion of great equal-
izer for some of our most vulnerable students. Given the results of this study and
others, it would appear that the most prudent approach to attaining such a goal is a
multifaceted one. A policy that moves students into a higher-level course at eighth
grade would likely have a much better chance for success if it were implemented
over multiple years, starting with changes in the curriculum prior to eighth grade.
Moreover, if policymakers find it imperative to implement such a policy at a given
point in the education path (i.e., eighth grade), an indispensable component should
be mandated additional support for students who do not handle the shift to higher
ground with the same proficiency.
In the final analysis, students in this study who were assigned to Algebra I in
eighth grade did not fare as well as students taking a lower mathematics course in
terms of 4-year college readiness by the eleventh grade, but readiness for 2-year and
selective 4-year colleges did not differ significantly. More importantly, neither
group fared well in comparison to the national average nor compared to those who
passed Algebra I. We submit that the best solution may not be to figure out where it
is best to place underprepared students, but rather to address the lack of prepared-
ness in a systematic manner throughout students’ elementary and middle school
experiences. To do otherwise would be to resign to the position that some students
cannot or will not acquire prerequisite knowledge by eighth grade that has been
proven to be instrumental to college access.
Acknowledgments
This research was supported by a grant from the American Educational Research Association, which
receives funds for its “AERA Grants Program” from the National Science Foundation under Grant
#DRL-0941014. Opinions reflect those of the authors and do not necessarily reflect those of the
granting agencies.
References
Acock, A. C. (2005). Working with missing values. Journal of Marriage and Family, 67(4), 1012–
1028.
Adelman, C. (2006). The toolbox revisited: Paths to degree completion from high school through
college. Washington, DC: U.S. Department of Education.
Allensworth, E., Nomi, T., Montgomery, N., & Lee, V. E. (2009). College preparatory curriculum
for all: Academic consequences of requiring algebra and English I for ninth graders in Chica-
go. Educational Evaluation and Policy Analysis, 31(4), 367–391.
Atanda, R. (1999). Do gatekeeper courses expand education options? (NCES 1999-303). Washing-
ton, DC: U.S. Department of Education, National Center for Education Statistics.
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 58
Attewell, P., & Domina, T. (2008). Raising the bar: Curriculum intensity and academic performance.
Educational Evaluation and Policy Analysis, 30(1), 51–71.
Boaler, J. (2011). Changing students’ lives through the de-tracking of urban mathematics class-
rooms. Journal of Urban Mathematics Education, 4(1), 7–14. Retrieved from
http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/view/138/85
Bowers, J., Fredrickson, M., & Hansen, B. (2010). RItools: Randomization inference tools. R pack-
age version 0.1-11. Retrieved from http://www.jakebowers.org/RItools.html
Cohen, J. W. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ:
Erlbaum.
College Board. (2012). Why community college. Retrieved from
http://professionals.collegeboard.com/guidance/college/community-college
Cremin, L. A. (1979). Horace Mann’s legacy. In L. A. Cremin (Ed.), The republic and the school:
Horace Mann on the education of free men. New York, NY: Teachers College Press. (Origi-
nal work published 1957)
Crocker, J., Major, B., Schmader, T., Spencer, S., & Wolfe, C. (1998). Coping with negative stereo-
types about intellectual performance: The role of psychological disengagement. Personality &
Social Psychology Bulletin, 24(1), 34–50.
Crocker, J., Major, B., & Steele, C. (1998). Social stigma. In D. T. Gilbert & S. T. Fiske (Eds.), The
handbook of social psychology (4th ed., Vol. II, pp. 504–553). Boston, MA: McGraw-Hill.
Dalton, B., Ingels, S. J., Downing, J., & Bozick, R. (2007). Advanced mathematics and science
coursetaking in the spring high school senior classes of 1982, 1992, and 2004 (NCES 2007-
312). Washington, DC: National Center for Educational Statistics, Institute of Education Sci-
ences.
Eccles, J. S., & Wigfield, A. (2002). Beliefs, values, and goals. Annual Review of Psychology, 53,
109–132.
Evan, A., Gray, T., & Olchefske, J. (2006). The gateway to student success in mathematics and sci-
ence: A call for middle school reform- The research and its implications. Washington, DC:
American Institutes for Research.
Faulkner, V. N., Stiff, L. V., Marshall, P. L., Nietfeld, J., & Crossland, C. L. (2014). Race and teach-
er evaluations as predictors of algebra placement. Journal for Research in Mathematics Edu-
cation, 45(3), 288–311.
Gamoran, A., & Hannigan, E. C. (2000). Algebra for everyone? Benefits of college preparatory math
students with diverse abilities in early secondary school. Educational Evaluation and Policy
Analysis, 22(3), 241–254.
Hansen, B., & Bowers, J. (2008). Covariate balance in simple, stratified and clustered comparative
studies. Statistical Science, 23(2), 219–236.
Ho, D. E., Imai, K., King, G., & Stuart, E. A. (2007). Matching as nonparametric preprocessing for
reducing model dependence in parametric casual inference. Political Analysis, 15, 199–236.
Howard, K. E. (2012). Hybrid technology classrooms for science and mathematics instruction. In T.
Swim, K. E. Howard, & I. H. Lim (Eds.), The hope for audacity: Recapturing optimism and
civility in education. New York, NY: Lang.
Howard, K. E., & Anderson, K. A. (2010). Stereotype threat in middle school: The effects of prior
performance on expectancy and test performance. Middle Grades Research Journal, 5(3),
119–137.
Howard, T. C. (2010). Why race and culture matter in schools: Closing the achievement gap in
America’s classrooms. New York, NY: Teachers College Press.
Iacus, S. M., King, G., & Porro, G. (2009). CEM: Coarsened exact matching software. Journal of
Statistical Software, 30, 1–27.
http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/view/138/85
http://www.jakebowers.org/RItools.html
http://professionals.collegeboard.com/guidance/college/community-college
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 59
Ingels, S. J., Dalton, B., Holder, T. E., Lauff, E., & Burns, L. J. (2011). High school longitudinal
study of 2009 (HSLS:09): A first look at fall 2009 ninth-graders. Washington, DC: National
Center for Education Statistics.
Ingels, S. J., Herget, D., Pratt, D. J., Dever, J., & Copello, E. (2010). High school longitudinal study
of 2009 (HSLS:09) Base-year field test report (NCES 2010-01). U.S. Department of Educa-
tion. Washington, DC: National Center for Education Statistics.
Ingels, S. J., Pratt, D. J., Herget, D. R., Dever, J. A., Fritch, L. B., Ottem, R., Rogers, J. E., Kitmitto,
S., & Leinwand, S. (2014). High school longitudinal study of 2009 (HSLS:09) Base year to
first follow-up data file documentation (NCES 2014-361). Washington, DC: National Center
for Education Statistics.
Keller, J. (2007). Stereotype threat in classroom settings: The interactive effect of domain identifica-
tion, task difficulty and stereotype threat on female students’ maths performance. British
Journal of Educational Psychology, 77(2), 323–338.
Learn.org. (2015). What are the admission requirements for junior and community colleges? Retrieved from
http://learn.org/articles/What_are_the_Admission_Requirements_for_Junior_Colleges_and_Community
_Colleges.html
Liang, J. H., Heckman, P. E., & Abedi, J. (2012). What do California standards test results reveal
about the movement toward eighth-grade algebra for all? Educational Evaluation and Policy
Analysis, 34(3), 328–343.
Loveless, T. (2008). The misplaced math student lost in eighth-grade algebra. (Special release: The
2008 Brown Center Report on American Education). Washington, DC: Brown Center on Ed-
ucational Policy.
Loveless, T. (2009). Tracking and detracking: High achievers in Massachusetts middle schools.
Washington, DC: Thomas B. Fordham Foundation.
Major, B., & O’Brien, L. T. (2005). The social psychology of stigma. Annual Review of Psychology,
56, 393–421.
Martin, D. B. (2003). Hidden assumptions and unaddressed questions in Mathematics for All rheto-
ric. The Mathematics Educator, 13(2), 7–21.
Mirel, J. (2006). The traditional high school: Historical debates over its nature and function. Educa-
tion Next, 6(1), 14–21.
Moses, R. P. (1995). Algebra, the new civil right. In C. B. Lacampagne, W. Blair, & J. Kaput (Eds.),
The algebra initiative colloquium (Vol. 2, pp. 53–67). Washington, DC: U.S. Department of
Education, Office of Educational Research and Improvement.
Moses, R. P., & Cobb, C. J. (2001). Organizing Algebra: The need to voice a demand. Social Policy,
31(4), 4–12.
Mosqueda, E. (2010). Compounding inequalities: English proficiency and tracking and their relation
to mathematics performance among Latina/o secondary school youth. Journal of Urban
Mathematics Education, 3(1), 57–81. Retrieved from
http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/view/47/48
National Center for Education Statistics. (2011). High school longitudinal study of 2009 (HSLS:09).
[Data file and code book]. Retrieved from http://nces.ed.gov/surveys/hsls09/
National Council of Teachers of Mathematics. (2008). Algebra: What, when, and for whom [Position
statement]. Reston, VA: National Council of Teachers of Mathematics.
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the na-
tional mathematics advisory panel. Washington, DC: U.S. Department of Education.
Nomi, T. (2012). The unintended consequences of an algebra-for-all policy on high-skill students:
Effects on instructional organization and students’ academic outcomes. Educational Evalua-
tion & Policy Analysis, 34(4), 489–505.
http://learn.org/articles/What_are_the_Admission_Requirements_for_Junior_Colleges_and_Community_Colleges.html
http://learn.org/articles/What_are_the_Admission_Requirements_for_Junior_Colleges_and_Community_Colleges.html
http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/view/47/48
http://nces.ed.gov/surveys/hsls09/
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 60
Nomi, T., & Allensworth, E. (2009). “Double-dose” algebra as an alternative strategy to remediation:
Effects on students’ academic outcomes. Journal of Research on Educational Effectiveness,
2(2), 111–148.
Nomi, T., & Allensworth, E. (2014). Skill-based sorting in the era of college prep for all: Costs and
benefits. University of Chicago Consortium on Chicago School Research. Retrieved from
http://ccsr.uchicago.edu/sites/default/files/publications/Sorting%20Brief.pdf
Oakes, J. (2005). Keeping track (2nd ed.). New Haven, CT: Yale University Press.
Paul, F. G. (2005). Grouping within Algebra I: A structural sieve with powerful effects for low-
income, minority, and immigrant students. Educational Policy, 19(2), 262–282.
Pintrich, P. R., & Schunk, D. H. (2002). Motivation in education (2nd ed.). New York, NY: Merrill
Prentice Hall.
Plunk, A. D., Tate, W. F., Bierut, L. J., & Grucza, R. A. (2014). Intended and unintended effects of
state-mandated high school science and mathematics course graduation requirements on edu-
cational attainment. Educational Researcher, 43(5), 230–241.
R Development Core Team (2008). R: A language and environment for statistical computing. Ven-
na, Austria: R Foundations for Statistical Computing.
Rosin, M., Barondess, H., & Leichty, J. (2009). Algebra in California: Great expectations and seri-
ous challenges. Mountain View, CA: EdSource.
Schmidt, W. H. (2004). A vision for a common, coherent, and challenging curriculum can transform
mathematics education in the United States. Educational Leadership, 61(5), 6–11.
Schmitz, B., & Perels, F. (2011). Self-monitoring of self-regulation during math homework behav-
iour using standardized diaries. Metacognition Learning, 6(3), 255–273.
Schneider, M. (2009). Math in American high schools: The delusion of rigor. Washington, DC:
American Enterprise Institute for Public Policy Research.
Schunk, D. H., Pintrich, P. R., & Meece, J. L. (2008). Motivation in education: Theory, research,
and applications. Upper Saddle River, NJ: Pearson.
Silva, C. M., Moses, R. P., Rivers, J., & Johnson, P. (1990). The algebra project: Making middle
school mathematics. The Journal of Negro Education, 59(3), 375–391.
Silver, D., Saunders, M., & Zarate, E. (2008). What factors predict high school graduation in the Los
Angeles unified school district? California Dropout Research Project Report. Santa Barbara,
CA.
Spencer, S. J., Steele, C. M., & Quinn, D. M. (1999). Stereotype threat and women’s math perfor-
mance. Journal of Experimental Social Psychology, 35, 4–28.
Spielhagan, F. R. (2006). Closing the achievement gap in math: The long-term effects of eighth-
grade algebra. Journal of Advanced Academics, 18(1), 34–59.
Steele, C. M. (1997). A threat in the air. American Psychologist, 52(6), 613–629.
Steele, C. M. (2003). Stereotype threat and African-American student achievement. In T. Perry, C.
M. Steele, & A. G. Hilliard, III, Young, gifted, and Black (pp. 109–130). Boston, MA: Bea-
con Press.
Steele, C. M., & Aronson, J. (1995). Stereotype threat and the intellectual test performance of Afri-
can Americans. Journal of Personality and Social Psychology, 69(5), 797–811.
Stein, M. K., Kaufman, J. H., Sherman, M., & Hillen, A. (2011). Algebra: A challenge at the cross-
roads of policy and practice. Review of Educational Research, 81(4), 453–492.
Sterner, W. R. (2011). What is missing in counseling research? Reporting missing data. Journal of
Counseling & Development, 89(1), 56–62.
Stone, C. (1998). Leveling the playing field: An urban school system examines equity in access to
mathematics curriculum. Urban Review, 30(4), 295–307.
Stuart, E. A. (2010). Matching methods for casual inference: A review and a look forward. Statistical
Science, 25(1), 1–21.
http://ccsr.uchicago.edu/sites/default/files/publications/Sorting%20Brief.pdf
Howard et al. Success after Failure
Journal of Urban Mathematics Education Vol. 8, No. 1 61
Téllez, K., Moschkovich, J., & Civil, M. (Eds.). (2011). Latinos/as and mathematics education: Re-
search on learning and teaching in classrooms and communities. Charlotte, NC: Information
Age.
Thoemmes, F., & Kim, E. S. (2011). A systematic review of propensity score methods in the social
sciences. Multivariate Behavioral Research, 46(1), 90–118.
Vriens, M., & Melton, E. (2002). Managing missing data. Marketing Research, 14(3), 12–17.
Walston, J., & McCarroll, J. C. (2010). Eighth-grade algebra: Findings from the eighth-grade round
of the early childhood longitudinal study, kindergarten class of 1998-99 (ECLS-K)(NCES
2010016). Washington, DC: Department of Education, National Center for Education Statis-
tics.
Waterman, S. (2010). Pathways report: Dead ends and wrong turns on the path through algebra.
Palo Alto, CA: Noyce Foundation.
Wigfield, A. (1994). Expectancy-value theory of achievement motivation: A developmental perspec-
tive. Educational Psychology Review, 6(1), 49–78.
Wigfield, A., & Eccles, J. S. (2000). Expectancy-value theory of achievement motivation. Contem-
porary Educational Psychology, 25(1), 68–81.
Williams, T., Haertel, E., Kirst, M. W., Rosin, M., & Perry, M. (2011). Preparation, placement, pro-
ficiency: Improving middle grades math performance. [Policy and practice brief]. Mountain
View, CA: EdSource.
Zinth, K. (2005). Student promotion/retention policies. Denver, CO: Education Commission of the
States.