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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] 123 172 ResHighEduc(2013)54:171–200 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). 123 ResHighEduc(2013)54:171–200 173 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 123 174 ResHighEduc(2013)54:171–200 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’’. 123 ResHighEduc(2013)54:171–200 175 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? 123 176 ResHighEduc(2013)54:171–200 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 123 ResHighEduc(2013)54:171–200 177 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. 123 178 ResHighEduc(2013)54:171–200 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 123 ResHighEduc(2013)54:171–200 179 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 123 180 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 123

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explaining the low rate of certificate completion among remedial math students who do not achieve college-level math analyses with a descriptive analysis of data from the National Center for Education Sta- Although the name applied to this credential varies from college to college, as do the spec
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