The Aftermath of Remedial Math

31 Pages·2013·0.5 MB·English

ResHighEduc(2013)54:171–200
DOI10.1007/s11162-012-9281-4
The Aftermath of Remedial Math: Investigating the Low
Rate of Certificate Completion among Remedial Math
Students
Peter Riley Bahr
Received:30June2012/Publishedonline:19January2013
(cid:2)SpringerScience+BusinessMediaNewYork2013
Abstract Nationally, a majority of community college students require remedial assis-
tance with mathematics, but comparatively few students who begin the remedial math
sequenceultimatelycompleteitandachievecollege-levelmathcompetency.Theacademic
outcomesofstudentswhobeginthesequencebutdonotcompleteitaredisproportionately
unfavorable: most students depart from the community college without a credential and
without transferring to a four-year institution. Interestingly, however, many of these stu-
dentscontinuetoattendthecommunitycollegeaftertheyexittheremedialmathsequence,
sometimesforanextendedperiod.Oneisledtoaskwhystudentswhodonotcompletethe
sequencegenerallyarenotfindingtheirwaytoanalternativecredentialobjectivethatdoes
not require college-level math competency, such as a career and technical education
certificate, sometimes referred to as a vocational certificate. In this study, I explore three
possible answers to this question, including difficulty navigating to the alternative cre-
dential, declining participation in the community college, and declining academic per-
formance. I find that all three of these explanations contribute (to varying degrees) to
explainingthelowrateofcertificatecompletionamongremedialmathstudentswhodonot
achieve college-level math competency.
Keywords Community College (cid:2) Remedial (cid:2) Developmental (cid:2) Math (cid:2) Vocational (cid:2)
Certificate
Introduction
Researchonremedialmathematicsincommunitycollegescontinuestorevealatroubling
state of affairs. Nationally, the majority of first-time community college students—about
two-thirds—requireremedialmathassistance(Baileyetal.2010).Yet,nearlythree-fourths
of the students who begin the remedial math sequence ultimately do not complete a
P.R.Bahr(&)
CenterfortheStudyofHigherandPostsecondaryEducation,UniversityofMichigan,School
ofEducation,Room2108,610E.UniversityAvenue,AnnArbor,MI48109-1259,USA
e-mail:[email protected]
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college-level math course successfully (Bahr 2012a; Bailey 2009; Fain 2012).1 Unfortu-
nately, students who begin the remedial math sequence but do not complete it are very
likely to leave college without a credential and without transferring to a four-year insti-
tution (Bahr 2008a, 2010a). Accordingly, Cullinane and Treisman (2010), citing compa-
rably troubling findings by Calcagno (2007) and Adelman (2004), describe remedial
educationas‘‘notanentrywaybutaburialgroundfortheaspirationsofmyriadcommunity
college students seeking to improve their lives through education’’ (p. 2).
Interestingly, however, the evidence indicates that students who drop out of the
remedialmathsequencewithoutachievingcollege-levelcompetencyoftenremainenrolled
inthecommunitycollegeaftertheirlastmathcourse,sometimesforanextendedperiodof
time (Bahr 2010b, 2012a; Perry et al. 2010). In other words, there is a seeming contra-
diction between the facts that (1) students who do not complete the remedial math
sequenceoftencontinueinthecommunitycollegeforanumberofsemestersaftertheyexit
thesequence,and(2)thesestudentsonlyrarelyleavethecommunitycollegewithanykind
ofcredential.Oneisledtoask,whatarestudentsdoingbetweentheirprematureexitfrom
the remedial math sequence and their eventual departure from the community college?
Whyarethese studentsnotfindingtheirwaytoacredentialprogramthathaslowermath
requirements, such as a career and technical education certificate (i.e., a vocational
certificate)?
Theobjectiveofthisstudyistobegintodeconstructhowstudentswhodropoutofthe
remedialmathsequencewithoutachievingcollege-levelcompetency,butremainenrolled
in the community college after their last math course, are using the community college
before and after their participation in math. Specifically, I seek to understand how the
course-taking behavior of these students in the period after their exit from the remedial
sequence (i.e., the after-math period) differs from their prior course-taking behavior and
from that of students who achieve college-level math competency. To accomplish this
objective, I use data from the California Community College (CCC) system to estimate
relationshipsbetweenselectedaspectsofstudents’course-takingbehaviorintheafter-math
period, their prior course-taking behavior, and their highest level of math achievement,
while controlling for other relevant variables. The focal course-taking behaviors, which
wereselectedbasedonreasonedinferencesfromtheliterature,includevocationalcourse-
taking, mean course credit load, and successful course completion. I supplement these
analyses with a descriptive analysis of data from the National Center for Education Sta-
tistics’ National Education Longitudinal Study (NELS) and Postsecondary Education
Transcript Study (PETS).
Background
Itisunquestionablytruethatassistingeverycommunitycollegestudenttoachievecollege-
levelmathcompetencyisahighlydesirablegoal,benefitingboththestudentsthemselves
and society as a whole. Consequently, remedial mathematics in community colleges has
been the subject of intensive research in the last decade, especially as it pertains to three
overarching questions that have framed much of the inquiry on the topic. One of these
1 Notethattherateatwhichremedialmathstudentsachievecollege-levelmathcompetencyvariesgreatly
bytheirpointofentrytothesequence:studentswhobeginthesequenceatlowerpointsofentry(greater
mathdeficiencies)haveasubstantiallylowerchanceofachievingcollege-levelmathcompetencythando
studentswhobeginathigherpointsofentry(Bahr2012a).
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questions concerns the factors that influence the successful attainment of college-level
mathcompetencyamongtheremedialmathstudentpopulationgenerally(e.g.,Bahr2007,
2008b,2010b)andamongspecificsubsetsofthatpopulation(e.g.,Bahr2010c).Asecond
questionaddressestheefficacyofmathematicsremediationalongavarietyofdimensions,
includingforexamplestudents’successinsubsequentcoursework(e.g.,Waycaster2001),
students’ long-term academic attainment (e.g., Attewell et al. 2006; Bahr 2008a, 2010a;
Bettinger and Long 2009; Melguizo et al. 2011), and the cost effectiveness of students’
participation in remediation (e.g., Melguizo et al. 2008). The third question concerns the
processofmathematicsremediation,particularlyhowstudentsprogressorfailtoprogress
through the steps of the remedial math sequence (e.g., Bahr 2009a, 2012a; Bailey et al.
2010; Hagedorn and DuBray 2010).
Thefindingsofthisresearchhavedriveneffortstodevise,implement,andtestmodels
ofmathematicsremediationandsupportinginterventionsthatwillimprovethedismalrate
at which students achieve college-level math competency and go on to complete post-
secondarycredentials.Workanddiscourseinthisregardhasfocusedonanumberoffacets
of mathematics remediation, such as the structure of remedial math curricula and the
methods of instruction (e.g., Biswas 2007; Bryk and Toch 2012; Cullinane and Treisman
2010; Grubb and Cox 2005; Grubb 2010; Hodges and Kennedy 2004; Klein and Wright
2009; Kozeracki 2005; Perin 2005; Perin and Charron 2006; Rutschow and Schneider
2011;SheldonandDurdella2010),basicskillstestingandcourseplacementpolicies(e.g.,
Bailey2009;Hadden2000),andinstitutionalsupportsforstudentpersistenceandsuccess
intheremedialmathsequence(e.g.,Bautsch2011;LevinandCalcagno2008;Perin2004;
Perry et al. 2010).
However,regardlessofthesuccessofcurrentorfutureeffortstoimprovetheremedial
math sequence, inevitably some students will begin the sequence but not complete it. At
present, this is the case for the majority of community college students who begin the
remedialmathsequence,anddisproportionatelythosewhobeginthesequenceatthelower
rungsoftheremedialladder(Bahr2012a;Baileyetal.2010).Thus,itperhapsissurprising
that, to date, very few (if any) studies have considered questions concerning how and to
what end remedial math students who do not achieve college-level math competency are
using the community college after exiting the remedial math sequence, and how we may
assist these students to identify alternative credentialing pathways with lower math
requirements (i.e., math requirements that are consistent with their level of attainment in
math)?
Onewouldhope,ofcourse,thatstudentswhodepartfromtheremedialmathsequence
withoutachievingcollege-levelmathcompetencywouldbeabletoidentifyandcomplete
an alternative goal that does not require college-level competency in math. In fact, most
community colleges offer an educational track that results in a marketable credential but
that usually does not require college-level math competency, namely the career and
technical education certificate, also known as the vocational certificate (Carnevale et al.
2012). Although the name applied to this credential varies from college to college, as do
the specific requirements for earning the credential, a certificate generally represents the
successful completion of a defined program of college coursework in a particular voca-
tionalfield.Unlikeanassociate’sdegree,however,acertificatetypicallydoesnotrequire
the completion of general education requirements, including college-level math and col-
lege-level English courses. As an example, California’s American River College offers
morethan120certificatesinfieldsasdiverseasautomotivetechnology,business,culinary
arts, database management, elder care, fashion design, and geographic information sys-
tems,tonamejustafew(AmericanRiverCollege2009).Althoughsomeofthecertificate
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programs require courses for which college-level math competency is an explicit pre-
requisite or recommended preparation, most do not. Hence, a certificate is a viable alter-
native credential for students who do not complete the remedial math sequence.
Theevidencesuggeststhatcertificateshavesignificantlabor-marketvalue(Belfieldand
Bailey 2011; Carnevale et al. 2012). For example, Dadgar and Weiss (2012), using data
fromWashington,foundthatlong-termcertificates—thoserequiringonefull-timeyearor
moretocomplete—producedgainsinearnings,ahigherlikelihoodofbeingemployed,and
more hours worked per week among employed individuals. The labor market returns of
short-term certificates—those requiring less than one-full-time year to complete—were
lesscompelling,butshort-termcertificatesincertainfieldsdidproducegainsinearnings.
Jepsenetal.(2009)foundsimilarresultsusingdatafromKentucky,thoughtheyfoundthat
even short-term certificates produce a higher likelihood of employment.2
Yet, by and large, the hope that remedial math students who do not complete the
remedialsequencewillleavethecommunitycollegewithamarketablecredential,suchas
a certificate, is not being realized.More than four-fifthsof these students depart from the
communitycollegewithoutacredentialandwithouttransferringtoafour yearinstitution
(Bahr 2008a, 2010a). Given that many remedial math students who do not complete a
college-level math course nevertheless persist in college after exiting the remedial
sequence(Bahr2010b,2012a;Perryetal.2010),oneisledtoaskwhythesestudentsare
not shifting their focus to, and ultimately earning, a certificate prior to departing from
college.Theextantliteratureprovidesanumberofhintsconcerningpotentialexplanations,
and I consider three likely possibilities in this study.
First,recentworkhasdrawnattentiontothedifficultyexperiencedbymanycommunity
collegestudentswithnavigatingthroughthecommunitycollege,whichisaresultinpartof
thegreatflexibilityofferedtostudentsintermsoftheacademicgoalsthattheypursueand
the timeline on which they pursue those goals (Bahr 2013; Moore and Shulock 2011;
Rosenbaum et al. 2009; Scott-Clayton 2011). In light of this observation, one might
suppose that students who depart from the remedial math sequence without achieving
college-levelmathcompetencysimplyarenotadjustingtheireducationalplanstoaccount
for the change in their level of math achievement and, consequently, not adjusting their
course-takingbehaviorintheafter-mathperiod.Inotherwords,themostlogicalplacefor
remedialmathstudentstodirecttheircourse-takingiftheydonotcompleteacollege-level
math course is vocational fields, but misunderstanding or confusion associated with nav-
igatingthecommunitycollegemaystallthisredirectionofacademiceffort.Ifstudentswho
donotcompletetheremedialmathsequencearenotmakingnecessaryadjustmentsintheir
academic plans, one would expect to find no change in their likelihood of enrolling in
vocational coursework during the after-math period relative to their prior participation in
vocational coursework.
Asasecond(thoughnotmutuallyexclusive)possibility,itisclearthatstudentsusethe
communitycollegeinavarietyofwaystomeetanarrayofends,andthesedifferinguses
of the community college are demonstrated through divergent course-taking and enroll-
ment patterns (e.g., Bahr 2010d, 2011). One conceptualization of variation in community
collegestudents’course-takingandenrollmentpatternsfocusesonstudents’accumulation
2 Jepsenetal.(2009)distinguishbetweenassociatedegrees,diplomas,andcertificates,incontrasttothe
distinctionmadebyDadgarandWeiss(2012)betweenassociatedegrees,long-termcertificates,andshort-
term certificates.Basedonthe definitions providedin eachpaper,the‘‘certificates’’ describedbyJepsen
etal.(2009)correspondtothe‘‘short-termcertificates’’describedbyDadgarandWeiss(2012),whilethe
‘‘diplomas’’correspondtothe‘‘long-termcertificates’’.
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ofcoursecreditovertime.Withinthisconceptualization,onecommonpatternidentifiedby
Marti (2008) is marked by a downward trend in semester-by-semester accumulation of
course credit—a group that he described as long-term decliners.
Asitpertainstothequestionsaddressedinthisstudy,itmaybethecasethatremedial
mathstudentswhoexittheremedialmathsequencewithoutachievingcollege-level math
competency are disproportionately likely to exhibit long-term declining behavior.
Although, on the face, this assertion may seem to be pure conjecture, in fact it could
explainboththefailuretoachievecollege-levelmathcompetencyandthefailuretoearna
credential prior to departing from college. That is, some students’ rate of progress (Bahr
2009a) may be slowing progressively as their time in the community college lengthens
untiltheyfirststoptakingmathcoursesandtheneventuallydropoutofcollegealtogether.
Ifthat were true,one would expect tofindthat the average course credit loadof students
who do not complete the remedial math sequence is significantly lower in the after-math
periodthanitwaspriortoexitingtheremedialmathsequence.Furthermore,adownward
trend in participation in the community college could negate the benefit of a shift to
vocational course-taking in the after-math period, if such a shift is occurring, preventing
even students who make the necessary adjustment in their academic trajectory from
completing a certificate prior to departing from college.
Finally,eitherseparatelyorinconjunctionwiththepreviouspropositions,itmaybethe
casethatstudentswhoexittheremedialmathsequencewithoutcompletingacollege-level
math course are disproportionately likely to be experiencing a downward trend in their
academicperformance.EvidenceofthispossibilitymaybefoundinBahr’s(2011)studyof
the relationship between students’ course-taking and enrollment patterns and their par-
ticipation in remedial math. In particular, Bahr found that nearly one quarter of remedial
math students in community colleges fit the course-taking and enrollment profile that he
described as experimental—students who typically enroll in a part-time course load and
whofailtocompletesuccessfullythemajorityoftheircoursesduringtheircomparatively
shorttimeinthecommunitycollege.Aswiththepossibilityofoverrepresentationoflong-
termdecliningbehavioramongunsuccessfulremedialmathstudents,overrepresentationof
declining academic performance could explain both the failure to complete the remedial
sequence and the fact that students who do not complete the remedial sequence are
unlikely to earn a credential prior to departing from college. If declining academic per-
formancewerepartofthereasonforthelowrateofcertificatecompletionamongremedial
math students who do not complete a college-level math course successfully, one would
expecttofindthatthesestudentshavealoweraveragerateofcoursesuccessintheafter-
mathperiod,relativetorateofcoursesuccesspriortoexitingtheremedialmathsequence.
Insum,Iconsiderthreepossibleexplanationsforthelowrateofcertificatecompletion
amongstudentswhoexittheremedialmathsequencewithoutachievingcollege-levelmath
competency, including difficulty navigating to the alternative vocational credential as
measured by vocational course enrollment rate, declining participation in the community
collegeasmeasuredbyaveragecoursecreditload,anddecliningacademicperformanceas
measured by course success rate. I test each of these possibilities by examining the rela-
tionship between students’ behavior after exiting the remedial math sequence (i.e., their
after-math behavior), their corresponding behavior prior to exiting the remedial math
sequence,andtheirhighestlevelofmathachievement.Ineachcase,Iseektoanswertwo
questions:
1. How does course-taking behavior in the after-math period differ from course-taking
behavior prior to exiting the remedial math sequence?
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2. Towhatextentdoesanyobservedchangeinafter-mathcourse-takingbehaviordepend
on how far through the remedial math sequence a student progressed prior to exiting
the sequence?
Data, Measures, and Methods
Data Sources
I draw on two sources of data to answer the questions posed in this study. The primary
source for this study is the system database maintained by the Chancellor’s Office of the
CCC. The CCC system database includes detailed transcript records, as well as demo-
graphicmeasures,financialrecords,recordsofcredentialawards,andthelike,forallofthe
communitycollegestudentsinCalifornia’s112communitycolleges.Thelargenumberof
student records available in these data makes them an excellent choice for the analytical
work of deconstructing students’ course-taking behavior before and after their exit from
the remedial math sequence, but caution must be exercised in generalizing findings to
community college systems in other states.
Given this limitation, I also draw on a secondary source: the National Educational
LongitudinalStudy(NELS:88/00)andtheassociatedPETS.Both studieswereconducted
by the National Center for Education Statistics (NCES) of the U.S. Department of Edu-
cation. The NELS data constitute a nationally representative sample of eighth-grade stu-
dentsinthespringof1988.Thesestudentsweresurveyedfirstin1988,andthenagainin
1990,1992,1994,and2000.ThecompanionPETSdataaddresstranscriptrecordsforeach
postsecondary institution attended.
I use the NELS/PETS data to estimate on a national scale how frequently community
college students exit the remedial math sequence without achieving college-level math
competency yet remain enrolled in the community college. Unfortunately, the small
numberofcommunitycollegestudentsintheNELS/PETSdatarestrictstheutilityofthese
data for the larger analytical purpose of deconstructing students’ course-taking behavior.
Analytical Cohorts
As it pertains to the CCC system database, I focus specifically on the fall 2002 first-
time student cohort in the 105 semester-based community colleges that were in oper-
ation in 2002. I defined this cohort to include students whose first term of enrollment in
for-credit coursework was the fall 2002, who were not concurrently enrolled in high
school in the fall 2002, who did not transfer college credit into the system at entry,
who did not have a postsecondary degree at entry, who had not enrolled in a four-year
institution within at least the previous 5 years, and who reported a valid social security
number. The last criterion (a valid social security number) was necessary to determine
the first term of for-credit enrollment in the system and to identify prior enrollments in
four-year institutions.
Thisresulting cohortiscomposed 190,637 students.Thecourse-taking and enrollment
behavior of these students was observed across the system through the summer term of
2010 and regardless of students’ decisions to transfer laterally between community col-
legesinthesystem(Bahr2009b,2012b).Ithenrestrictedthecohorttothosestudentswho
enrolledinthesystemforatleastonesemesteraftertheirfirstsemesterofattendanceinthe
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fall2002andwhosefirstnon-vocationalmathcoursewasremedialinnature,resultingina
final analytical cohort of 79,545 students.
The analytical cohort from the NELS/PETS data was defined to parallel the cohort
selected from the CCC system database, focusing on the fall 1992 first-time students in
semester-basedcommunitycollegeswhosefirstnon-vocationalmathcoursewasremedial
in nature and who enrolled in the community college for at least one semester after their
firstsemesterofattendance.ThefinalanalyticalcohortfortheNELS/PETSdataincludes
570 students whose course-taking and enrollment behavior was observed through the
spring term of 2001.3
Analytical Periods
Thefocusofthisstudyischangeinstudents’course-takingbehaviorintheperiodoftime
after they exit the remedial math sequence—the after-math period—relative to their
behaviorbetweencollegeentryandremedialmathexit.FollowingBahr’s(2012a)work,I
definedremedialmathtoincludeallnon-vocationalmathcoursesofaskill-levellessthan
college algebra. In order from lowest skill to highest skill, remedial math includes arith-
metic, pre-algebra, beginning algebra, intermediate algebra, and geometry, of which the
latter two are of equivalent relative skill. College-level math includes college algebra,
statistics,trigonometry,andallmathcoursesofgreaterskill,suchaspre-calculus,calculus,
differential equations, linear algebra, and the like.
Thesemesterinwhichastudentexitstheremedialmathsequenceisoperationalizedas
the semester of the first successfully completed college-level math course or, in the
absence of a successfully completed college-level math course, the semester of the last
attempted remedial or college-level math course. A successful math course enrollment is
one resulting in a grade of A, B, C, Credit, or Pass.
In some cases, the semester of a student’s exit from the remedial math sequence also
was the student’s last semester of enrollment in the community college system. By defi-
nition,suchstudentsdidnothaveanafter-mathperiodand,therefore,wereexcludedfrom
certainanalysesinthisstudy.Italsoispossiblethatthesemesterofastudent’sexitfrom
the remedial math sequence corresponds with the last semester observed in the data
(summer 2010 for the CCC data; spring 2001 for the NELS/PETS data), which is a
problemofcensoring.However,thiscorrespondenceoccurredonlyveryrarelyduetothe
lengthy observation period for both of the analytical cohorts employed in this study
(10 yearsfortheCCCanalyticalcohort;nineyearsfortheNELS/PETSanalyticalcohort).
Dependent Variables
I constructed three dependent variables for this study. The first two are course-level
variablesaddressingwhetheragivenfor-creditcoursetakenbyastudentintheafter-math
period was vocational in nature and, separately, whether the course was completed suc-
cessfully(grade of A, B, C, Credit,or Pass).To constructthese variables, Iidentified the
firstsemesterofenrollmentfollowingexitfromtheremedialmathsequenceforallstudents
who continued in the community college after exiting remedial math. I then coded all
coursesthatweretakeninthissemesterorasubsequentsemesterasvocational(= 1)ornot
(= 0) and,separately, as completedsuccessfully (= 1) or not (= 0). Thevocational nature
of a course was determined using the six-digit system in California’s Taxonomy of
3 Thereportedsamplesizewasroundedtothenearestten,perNCESguidelines.
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Programs(Chancellor’sOffice2009).Forthepurposeofdeterminingwhetheracoursewas
completed successfully, grade records of Withdrawal were treated as unsuccessful out-
comes and coded as zeros.
The third dependent variable is mean course credit load in regular semesters after
exiting the remedial math sequence. I constructed this variable by calculating the total
numberofcreditsattemptedbyastudentinfallandspringsemestersduringtheafter-math
period,andthendividingthistotalbythenumberoffallandspringsemestersintheafter-
mathperiodinwhichastudentenrolledinanycoursework.Summertermswereexcluded
because even full-time students tend to enroll in lower course credit loads during these
terms,leadingtodepressedvaluesonthismeasurerelativetostudentswhodidnotenrollin
coursework during the summer.
Independent Variables
The focal independent variables in this study are similar to the dependent variables but
addresstheperiodoftimebetweencollegeentryandremedialmathexit,asopposedtothe
period of time after exiting remedial math. The first is the proportion of non-math, for-
credit courses taken by a student during this period that were vocational in nature. This
measurewascalculatedbyidentifyingthesemesterinwhichastudentexitedtheremedial
math sequence, dropping all courses that were attempted by the student after exiting the
remedial math sequence, dropping all remedial and college-level math courses that were
attemptedbythestudent,andthencalculatingtheproportionoftheremainingcoursesthat
were vocational in nature.
Mathcourseswereexcludedfromthecalculationofthisindependentvariabletoavoid
confoundingvocationalcourse-takingwithcoursecreditload.Asahypotheticalexample,
consider that students who consistentlyattempted a half-time (six-credit) course loadand
consistentlyenrolledinmathrarelywouldhaveaproportionofvocationalcoursesgreater
than 0.5 (i.e., one three-credit math course and one three-credit vocational course), while
studentswhoenrolledinafull-time(twelve-credit)loadcouldhaveagreaterproportionof
vocational courses simply due to their greater course load (e.g., one three-credit math
course and three three-credit vocational courses). Eliminating math courses from the
calculationofthisvariableallowedbothpart-timeandfull-timestudentstohaveapossible
range of zero to one.
Thesecondofthefocalindependentvariablesistheproportionofnon-math,for-credit
coursesattemptedbyastudentintheperiodbetweencollegeentryandremedialmathexit
thatwerecompletedsuccessfully.Thismeasurewascalculatedinthesamemannerasthe
proportion of vocational courses with the exception that it addresses a successful grade
outcome rather than the vocational nature of the course.
Thelastofthefocalindependentvariablesismeancoursecreditloadinregular(falland
spring)semestersintheperiodbetweencollegeentryandremedialmathexit.Thismeasure
wascalculatedinamannercomparabletothatofmeancoursecreditloadintheafter-math
period. Note that, unlike the proportion of vocational courses and proportion of courses
completedsuccessfully,thecalculationofthisvariableincludesremedialandcollege-level
mathcoursesinwhichastudentenrolledintheperiodbetweencollegeentryandremedial
math exit.
In addition to the three focal independent variables, I incorporate into the analyses a
student’s point of entry to the remedial math sequence, point of exit from the remedial
mathsequence,demonstratedwritingcompetencyatexitfromtheremedialmathsequence,
self-reported academic goal at college entry, and whether or not a student completed a
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credentialduringtheperiodoftimebetweencollegeentryandremedialmathexit.Pointof
entry to the remedial math sequence was operationalized as the skill-level of a student’s
first remedial math course, while point of exit was operationalized as the skill-level of a
student’s highest-skill, successfully-completed math course, if any. Similarly, writing
competencyatexitfromtheremedialmathsequencewasoperationalizedastheskill-level
ofastudent’shighest-skill,successfully-completedwritingcourse(Bahr2012a)duringthe
periodoftimebetweencollegeentryandremedialmathexit,ifany.Thecompletionofa
credentialpriortoexitingremedialmathincludesbothcertificatesandassociatedegrees.A
totalof5,735students(7.2 %)intheCCCanalyticalcohortcompletedacredentialpriorto
exiting the remedial math sequence.
Methods of Analysis
I employed two distinct statistical methods in this study, depending on the nature of the
focal dependent variable. In the case of the two course-level, dichotomous dependent
variables,Iusedarandom-intercept(multilevel)logisticregressionmodel(Rabe-Hesketh
andSkrondal2008)inwhichcourseswerenestedwithinstudents.Thedependentvariable
thataddresseswhetheragivencourseattemptedintheafter-mathperiodwasvocationalor
not was regressed on the proportion of non-math courses attempted by a student in the
period prior to exiting the remedial math sequence that were vocational in nature, the
square root of the proportion of vocational courses prior to exiting the remedial math
sequence,pointofexitfromtheremedialmathsequence,themultiplicativeinteractionof
proportionofvocationalcourses(bothidentityandsquareroot)andpointofexit,pointof
entry to the remedial math sequence, the highest-level writing course completed suc-
cessfullybyastudentpriortoexitingtheremedialmathsequence,student’sself-reported
academicgoalatcollegeentry,andwhetherornotastudentcompletedacredentialpriorto
exitingtheremedialmathsequence.Themultiplicativeinteractionswereincludedbecause
mysecondresearchquestionaskstowhatextentchangesinastudent’safter-mathbehavior
depend on how far through the remedial math sequence the student progressed prior to
exiting the sequence.
TheequationusedhereisshownasEQ1.Theleft-handsideoftheequationrepresents
theloggedoddsoftheprobabilitythataparticularcourse(i)takenbyagivenstudent(j)in
theafter-mathperiodwasvocationalornot.Thisoutcomeisassumedtobeafunctionofk
observedfixedeffects(representedbyb ),includingtheinteractions,aconstant(b ),anda
k 0
randomeffectf,whichcapturesunobserveddifferencesbetweenstudentswithrespectto
j
theirprobabilityof experiencingapositive conditiononthedichotomousdependentvari-
able(Rabe-HeskethandSkrondal2008).Therandomeffectisanimportantaspectofthis
modelbecauseonewouldexpectthattheoutcomesexperiencedbyanygivenstudentare
morealikethanaretheoutcomesexperiencedbyanytwostudentschosenatrandom(Bahr
2009a).Further,thenumberofoutcomescontributedtothemodelbyeachstudentvaried,
resultinginsomestudentshavingmore‘‘weight’’intheestimationofthemodelbyvirtueof
attempting a greater number of courses. Hence, the random effect serves the purpose of
correctingthedifferentialweightonthemodelofunobservedstudent-specifictendenciesto
experienceapositiveconditiononthedependentvariable.Notethatalloftheindependent
variablesinthismodelaremeasuredatthestudent-level,asopposedtothecourse-level.
n (cid:2) (cid:3)o
logit P y = 1jx ;f ¼b þb x þf ð1Þ
ij kj j 0 k kj j
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180 ResHighEduc(2013)54:171–200
Table1 Percentage of students who continued in the CCC system after exiting the remedial math
sequence,bypointofentrytoandpointofexitfromtheremedialmathsequence(N=79,545)
Pointofexitfrommath
Pointof College- Intermediate Beginning Pre- Basic Noneor All
entryto level algebraor algebra algebra arithmetic vocational points
math math planegeom. mathonly ofexit
Interm.algebraor 90% 73% 66% 45% 69% 74% 82%
planegeom. (9,332) (3,913) (313) (73) (29) (3,883) (17,543)
Beginningalgebra 86% 67% 72% 51% 54% 69% 75%
(7,979) (3,716) (7,071) (282) (140) (7,094) (26,282)
Pre-algebra 82% 65% 67% 63% 60% 67% 68%
(3,033) (1,524) (3,109) (4,614) (205) (4,933) (17,418)
Basicarithmetic 83% 63% 69% 57% 64% 65% 66%
(1,777) (989) (2,284) (1,828) (5,462) (5,962) (18,302)
Allpointsofentry 87% 69% 70% 61% 63% 69% 73%
(22,121) (10,142) (12,777) (6,797) (5,836) (21,872) (79,545)
The dichotomous dependent variable that addresses whether a given course attempted
byastudentintheafter-mathperiodwascompletedsuccessfullyornotwasanalyzedinthe
samemannerasthedependentvariablethataddresseswhetheragivencourseattemptedby
a student was vocational or not, except for the substitution of the corresponding inde-
pendent variable and interactions. The dependent variable that addresses mean course
credit load in the after-math period was analyzed using a simpler ordinary least squares
linearregressionmodel(i.e.,notamultilevelmodel).Otherwise,though,theanalysiswas
handled inthesame manneras the twodichotomous dependentvariables,usingthe same
set of independent variables.
Analysis
Continuing in the Community College
To measure the frequency with which students exit the remedial math sequence without
achievingcollege-levelmathcompetencyyetcontinueinthecommunitycollege,Ipresent
inTables 1and2thepercentageofstudentswhocontinuedinthecommunitycollegeafter
exiting remedial math by point of entry to and point of exit from the remedial math
sequence. Table 1 provides percentages for the CCC system, while Table 2 provides
national estimates from NELS:88/00 and the associated PETS data. Recall that both
analyticalcohortswereconfinedtostudentswhoenrolledinthecommunitycollegeforat
least one semester after their first semester of attendance. Note also that small absolute
(unweighted) cell sizes required that students in the NELS data who entered/exited the
remedial math sequence at the two lowest points be collapsed into a single category.
OneobservesinTables 1and2thatamajorityofstudentswhoexitedtheremedialmath
sequence without achieving college-level math competency continued to attend college:
two-thirds (68 %) of students in California and three-fifths (60 %) of students nationally.
The high prevalence of this pattern of behavior holds for all points of entry to remedial
math,frombasicarithmetic(66 %ofstudentsinCalifornia;60 %ofstudentsnationally)to
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