121
ActiveLearningMathematicsatthe
UniversityofNebraska-Lincoln
WendyM.Smith
UniversityofNebraska-Lincoln
[email protected]
OneofthetoptwogoalsoftheMathTeacherEducationPartnership(MTE-P)isto
increasethequantityofsecondarymathematicsteacherspreparedatthepartnerinstitutions
by40%.Suchanincreasenecessarilyinvolvesmoreundergraduatesinterestedinmathematics.
Typically,beforeundergraduatesbecomemathematicsmajors,theytakeoneormore
freshman-levelmathematicscourses(precalculustocalculus2).However,in1987
mathematiciansreportedthat“asmanyas40%ofundergraduateswerefailingintroductory
calculus,andeventhosewhopasseddidnotappreciatethesubject’srelevance”(Wilson,1997,
p.A12).Morerecently,theCharacteristicsofSuccessfulProgramsinCollegeCalculus
(Bressoud,Carlson,Mesa,&Rasmussen,2013)showedfailurerates(gradesofD,For
Withdraw)rangedfromanaverageof25%atPh.D.-grantinguniversitiestoanaverageof37%
atregionalcomprehensiveuniversities.Thus,improvingstudents’experiencesinfreshmanlevelmathematicscoursesisalogicalprerequisitetoincreasingthenumberofundergraduates
whoseektobecomemathematicsteachers.
Inrecentyears,researchoverwhelminglyhasshownthatactivelearningstrategiesare
effectiveatincreasingstudentsuccessinundergraduatescienceandmathematicscourses(e.g.,
Bressoud&Rasmussen,2015;Freemanetal.,2013;Laursenetal.,2011;2014),particularlyfor
underrepresentedgroupsofstudents.Thus,theMTE-Pchosetofocusoneofitsfiveinitial
ResearchActionClustersaroundActiveLearningMathematics(ALMRAC).TheDepartmentof
MathematicsattheUniversityofNebraska-Lincoln(UNL)wasoneofthefoundingmembersof
thisALMRAC,havingalreadybeguneffortstoimproveprecalculuscourses.Themathematics
departmentalsoembracedtheMTE-PNetworkedImprovementCommunity(NIC;Bryketal.,
2015;Martin&Gobstein,2015)model.WorkingaspartoftheALMRAChasallowedUNLto
acceleratecourseimprovementsbybuildingonthelearningofotherALMRACinstitutions.
Further,thepositiveresultsofimplementingactivelearninginprecalculushasinstitutionalized
thesepracticesatUNL.
UNLutilizestheALMRACdefinitionofactivelearning:WedefineALMasteaching
methodsandclassroomnormsthatpromote:(1)students’deepengagementinmathematical
reasoning,(2)peer-to-peerinteraction,and(3)instructorinquiryintostudentthinking.In
Lawler,B.R.,Ronau,R.N.,&Mohr-Schroeder,M.J.(Eds.).(2016).ProceedingsofthefifthannualMathematics
TeacherEducationPartnershipconference.Washington,DC:AssociationofPublicLand-grantUniversities.
122
support,thecurriculumshouldfocusonkeymathematicalideas,includingtheuseoftasksthat
promotesensemakingandproceduralfluency.Studentactivityshouldfavoropportunitiesfor
themtoproposequestions,communicatereasoning,andsharesolutionsinprocess.Instructor
activityshouldshowcasepracticesthatpromotestudentengagementandbuildonstudent
thinkingtoadvancethemathematicalagenda.Thisclassroomdescriptionisrepresentedin
Figure1.
Figure1.Classroom-levelALMfeatures.
Adoptinganactivelearningapproachtoinstructionisonlyonecomponentofalarger
institutionaltransformationtoimprovestudentsuccessinmathematics.Besidesfaculty
leadership,activecampusanddepartmentalleadershiparealsocriticaltosuccessful
institutionalizationofALM.Departmentscommittedtoinstitutionaltransformationattendto
studentdata,particularlyplacementdata,studenttrajectoriesthroughcourses,andstudent
success.Inadditiontoeffectiveleadershipandintellectualresources,physicalresourcessuch
asclassroomfurnitureandclassminutesperweekalsocancontributetoorhinderinstitutional
transformation—representedinFigure2.UNLhascommittedtoinstitutionalchangeatthe
departmentalandclassroomlevels.Theoverallpurposeofthisbriefreportistosummarizethe
UNLeffortstotransformfreshman-levelmathematicscoursesandraisestudentsuccess.
Background
TheresearchteamatUNLcollectsextensivedataeachsemesteraboutthestudentsin
theprecalculusandcalculuscourses.Thisbriefreportfocusesondocumentingtheinstitutional
changesatNebraska,drawingmainlyonthequantitativedatacollectedaboutstudentsand
courses.UNLjoinedtheBigTenin2011,andstartedtofocusonraisingthesix-yeargraduation
ratesandfreshmanretentionrates.AtUNL,overtwo-thirdsoffreshmentakeamathematics
courseintheirfirstsemester;nootherdepartmentevengarnershalfoffirst-timefreshmen
Lawler,B.R.,Ronau,R.N.,&Mohr-Schroeder,M.J.(Eds.).(2016).ProceedingsofthefifthannualMathematics
TeacherEducationPartnershipconference.Washington,DC:AssociationofPublicLand-grantUniversities.
123
enrollment.Thus,freshmenretentioncorrelatesveryhighlywithpassingprecalculusand
calculusmathematicscourses.
Figure2.Department-levelALMfeatures.
AtUNL,fallenrollmentinmathematicscoursestendstobeapproximatelydoublethat
ofspringenrollment.InFall2015,approximately1,500studentstookoneoffourtypesof
precalculuscourses,andapproximately2,000studentstookcalculus1or2.Precalculuscourses
atUNLinclude:IntermediateAlgebra(3credithoursbutdoesnotcountasmathematics
credits),CollegeAlgebra(3credithours),Trigonometry(2credithours),andCollegeAlgebra
&Trigonometry(5credithours;theunionoftheprevioustwocourseslisted).
Table1
RelevantparametersofActiveLearningMathematicscontextatUNL
UNL–BytheNumbers(Fall2015)
AllStudents
Undergrads
FirstTimeFreshmenAverageACTScore
University-wideFreshman/SophomoreRetention
6-yeargraduation
MathGTAs
MathTenure-trackFaculty
25,260
20,182
25.2
82.5%
67.0%
77
29
Lawler,B.R.,Ronau,R.N.,&Mohr-Schroeder,M.J.(Eds.).(2016).ProceedingsofthefifthannualMathematics
TeacherEducationPartnershipconference.Washington,DC:AssociationofPublicLand-grantUniversities.
124
MethodstoImproveFreshmanSuccessinMathematics
Withsubstantialvariationacrosssections,historicalsuccessrates1inCollegeAlgebra
andCollegeAlgebra&Trigonometry(2007-2011)hoveredinthelow-60%range.Thefirst
effortstoimprovethesecoursesinvolvedanemporiummodel,inwhichstudentsworkedat
theirownpacethroughmaterialsinacomputerlab,withgraduateteachingassistants(GTAs)
availabletoanswerquestions.Thatmodelwasquicklyabandonedin2012,atwhichtimethe
focusshiftedtoactivelearningapproaches.
UNLinvestedconsiderableresourcesintoimplementingactivelearning,addressingallof
thecomponentsinFigure2.Successratesnowaresteadyataround80%.Keyeffortsincluded
First-YearMathematicsTaskForce,hiringaPrecalculusCoordinator,strongadministrative
leadershipandphysicalresources,theuseofLearningAssistants,acoursereadinessactivity,
andanefforttoimprovestudents’onlinehomeworkexperience.Eachisdiscussednext.
TheFirst-YearMathematicsTaskForcewasafacultycommitteededicatedtoimproving
freshman-levelmathematicscourses;akeyrolewastoevaluatethequalityanddifficultyofthe
exams,toprovideassurancethatimprovedsuccessrateswerenotduetoloweringof
mathematicalstandards.Thetaskforcealsomarshalledandmadesenseoflocaldata,tohelp
informtheongoingefforts.
UNLhiredafulltimeassistantprofessorofpracticetoserveascoordinatorforthefour
precalculuscoursesandtooverseethetrainingandongoingmentoringofGTAs.Close
coordinationofthecoursesincludesnotjustasyllabusandcommonexams,butalsodetailed
lessonplans,toreinforceamodelofclasstimedominatedbystudentsdoingmathematicsin
groups,andengaginginsmall-groupandwhole-classdiscussionstomakesenseofthe
mathematics.Thefirst-yearGTAsareresponsibleforteachingtheirowncourse(insectionsof
40students),andtheyarerequiredtotakeayear-longteachingseminar.Tobalancethis
demand,theGTAs’first-yearteachingassignmentisonecourseeachsemester;inlateryears,
GTAsteachtwocoursesinthefallandoneinthespring.
Duetostrongadministrativeleadership,multipleclassroomsoncampuswere
renovatedtoincludemoveabletablesandchairs,andwhiteboardsallaroundtheroom,and
thendedicatedtohousemathcourses.Additionally,understandingthatactivelearningtakes
moretimethanlecture,whilethecredithoursremainedconstant,theminutesperweekin
CollegeAlgebrawereincreasedfrom150to225,andtheminutesperweekinCollegeAlgebra
&Trigonometrywereincreasedfrom250to300.
AfinaleffortwastheimplementationofLearningAssistantsintotheprecalculusand
calculusmathematicscourses.Thedepartmenthiresundergraduatelearningassistants,initially
1
SuccessheremeansthestudentearnedaCorbetterinthecourse(orPassiftakenpass/nopass).
Lawler,B.R.,Ronau,R.N.,&Mohr-Schroeder,M.J.(Eds.).(2016).ProceedingsofthefifthannualMathematics
TeacherEducationPartnershipconference.Washington,DC:AssociationofPublicLand-grantUniversities.
125
grant-funded,tohelpintheprecalculuscoursesandtoserveasanadditionalinstructorto
facilitategroupdiscussions.
Inadditiontotheaboveactivities,theDepartmentofMathematicsalsodecidedto
instituteatestonthefirstdayofclass,coveringprerequisitematerial,andwhichcanserveasa
checkoncourseplacement.Thedepartmentquicklylearnedtocallthisa“coursereadiness
activity”ratherthan“prerequisitemasteryexam”;studentswhodidnotpassonthefirstday
wereallowedtoretakethetestonceperdayfortwoweeksinthecollegetestingcenter.
StudentscouldgetreviewpacketsandworkontheseintheMathResourceCenter(tutoring
roomstaffedbyGTAsandundergradmathmajors)withtheirinstructororlearningassistant.
Thiscoursereadinessactivitysortedstudentsintothreegroups:thosewhopassedon
thefirstday(knowthematerial);thosewhoeventuallypassed(investedtimetorelearn
material);thosewhoneverpassed(noobvioustimeinvestment).Mostofthe“neverpassed”
groupneverattemptedtotakethetestinthetestingcenteratall.Thiscoursereadinessactivity
hasprovedtobeincrediblypredictive.Thoseinthefirsttwogroups(pass/eventuallypass)
average80%onthecourseexams(threeexams+final),with75%ofstudentsinthosegroups
earninggradesof75%orhigher.Studentsinthe“neverpassed”groupaveragedunder60%on
courseexams,withnearly75%ofthesestudentsearninggradesbelowaC.Amongthestudents
whodonotpassthecoursereadinessactivityonDayOne,thechancestoretakethetestseem
tohelpdistinguishstudents’“collegereadiness”skillsofbeingabletoengageinpositivehabits
suchasstudyingandseekinghelp.
Oncethemathematicsdepartmentrealizedthepredictiveabilityofthiscourse
readinessactivity(Fall2014),thedepartmentsoughttoimplementinterventionswiththe
“neverpassed”groupinFall2015.Theinterventionstargetedmathskills,andweresuccessful
inraisingstudentexamaverageshigherthanthepreviousyears’“neverpassed”group,butnot
tothelevelofthosestudentswhodidpassthisinitialtest.Futureinterventionsmayneedto
findawaytoincorporatebroader“collegereadiness”skills.
Finally,aninternalgrantallowedtwofacultymemberstoimprovetheonlinehomework
experienceforstudents.UNLusesWeBWorK,anopen-sourceplatform,todeliveronline
homeworkassignmentstostudents,asawaytoprovidestudentsopportunitiestopractice
whatislearnedinclass.WeBWorKwasadoptedin2013,followingtheprevioususeof
MyMathLab.Studentsgetmultipleattemptsperitem;theonlinesystemallowsstudentsto
knowimmediatelyiftheiranswersarecorrect.Yet,knowingananswerisincorrectalmost
neverhelpsastudentdeterminehowtocorrectlysolveaproblem.
StudentswereinitiallyfrustratedthatWeBWorKlackedthe“hint”featureof
MyMathLab,inwhichastudentcouldseethesameproblem(withdifferentnumbers)worked
outinastep-by-stepfashion.However,thistypeofhintemphasizesprocedurememorization,
Lawler,B.R.,Ronau,R.N.,&Mohr-Schroeder,M.J.(Eds.).(2016).ProceedingsofthefifthannualMathematics
TeacherEducationPartnershipconference.Washington,DC:AssociationofPublicLand-grantUniversities.
126
andthedepartmentwantedtoemphasizeconceptualunderstanding.Thus,throughthis
internalgrant,experiencedGTAsandseveralhighschoolteacherswerehiredtocreatemore
conceptualhintstotargetcommonerrorsormisconceptions.Thesenewhintsweredesigned
tobethequestionaninstructorwouldask,orareminderaninstructorwouldgivetostudents
whowerestuckonaproblem,suchastheoneshowninFigure3.
Figure3.SampleimprovedhintofferedintheWeBWorkenvironment.
WeBWorKallowsadepartmenttokeeptrackofproblem-leveldata,sotheDepartment
ofMathematicscouldtrackthequestionswithandwithouthints.Studentsearnedmorepoints,
onaverage,onhomeworkquestionsthatincludedthesenewhints.Inresponsetoanend-ofsemestersurvey,84%ofstudentsreportedusingthehintsatleastonce,and87%ofthose
studentsfoundthehintstobeatleastoccasionallyhelpful,demonstratedinFigure4.In
craftinghints,thegoalwastoaddressthemostcommonmisconceptions,knowingthatthe
hintswouldnotbeabletocomprehensivelyaddresseverypossiblestudenterror.
Fall2015Percentage
100
90
80
70
60
50
40
30
20
10
0
OverallHomework HomeworkMean: HomeworkMean:
Mean
Hintproblems
NoHints
Figure4.Resultsofstudentsurveyshowingthedifferenceinhomeworkscoresonproblemswithand
withouthints.
Lawler,B.R.,Ronau,R.N.,&Mohr-Schroeder,M.J.(Eds.).(2016).ProceedingsofthefifthannualMathematics
TeacherEducationPartnershipconference.Washington,DC:AssociationofPublicLand-grantUniversities.
127
Conclusion
TheresearchteamatUNLcontinuestobeinvolvedintheALMRAC.Withactivelearning
well-established(successfully)inprecalculuscourses,thefocusin2016isonextendingactive
learningstrategiesintocalculus1andcalculus2courses.Inthosecourses,taughtinlarge(150)
lecturewithsmall(25-30)recitations,thelecturesarenowinfusedwithnumerousclicker
questions,whichtheinstructorcanfollowupbyhavingstudentsdiscusswithstudentsnearby.
Therecitationshaveshiftedfrom100to150minutesperweek,andinsteadofservingastimes
whenstudentswatchaGTAsolvehomeworkproblems,nowtheserecitationsserveastimefor
studentstoworkcollaborativelyonnewproblems.
TheWeBWorKhintsareexpandingtoothercourses(initialfundingfocusedonCollege
Algebra).Afuturegoalistocreatesomeshortvideohints,inwhichaninstructorgivesavery
brief(lessthan3minutes)lectureaboutthetopic,coveringmoreofthecommon
misconceptions.
Finally,theUNLresearchteamispartofanewgrantfromtheNationalScience
Foundation,titledSEMINAL:StudentEngagementinMathematicsthroughanInstitutional
NetworkforActiveLearning.SEMINALseekstostudytransformeddepartmentslikethoseat
UNL,UniversityofColorado-Boulder,andSanDiegoStateUniversity,aswellasdepartmentsat
earlierstagesoftransformation,bothinsideandoutsidetheALMRAC.Byfocusingonthe
NetworkedImprovementCommunitymodel,whatSEMINALlearnsaboutdepartmental
transformationcanberapidlydisseminatedtotherestoftheALMRAC,aswellastotheMTE-P
morebroadly,tosupportthepropagationofimprovedstudentsuccess.
ForMoreInformation
•
URL–www.math.unl.edu
•
Contact–WendySmith,UniversityofNebraska-Lincoln,[email protected]
References
Bressoud,D.,Carlson,M.P.,Mesa,V.,&Rasmussen,C.(2013).Thecalculusstudent:Insightsfromthe
MathematicalAssociationofAmericanationalstudy.InternationalJournalofMathematical
EducationinScience&Technology,44(5),685-698.
Bressoud,D.,&Rasmussen,C.(2015).Sevencharacteristicsofsuccessfulcalculusprograms.Noticesof
theAmericanMathematicalSociety,62(2),144-146.
Bryk,A.S.,Gomez,L.,Grunow,Al.,&LeMahieu,P.(2015).Learningtoimprove:HowAmerica’sschools
cangetbetteratgettingbetter.Boston:HarvardEducationPublishing.
Freeman,S.,Eddy,S.L.,McDonough,M.,Smith,M.K.,Okoroafor,N.,Jordt,H.,&Wenderoth,M.P.
(2014).Activelearningincreasesstudentperformanceinscience,engineering,andmathematics.
Lawler,B.R.,Ronau,R.N.,&Mohr-Schroeder,M.J.(Eds.).(2016).ProceedingsofthefifthannualMathematics
TeacherEducationPartnershipconference.Washington,DC:AssociationofPublicLand-grantUniversities.
128
ProceedingsoftheNationalAcademyofSciencesoftheUnitedStatesofAmerica(PNAS),111(23).
Retrievedfromwww.pnas.org/cgi/doi/10.1073/pnas.1319030111
Kezar,A.,&Gehrke,S.(2015).CommunitiesoftransformationandtheirworkscalingSTEMreform.Los
Angeles,CA:PulliasCenterforHigherEducation.Retrievedfrom:www.uscrossier.org/pullias/wpcontent/uploads/2016/01/communities-of-trans.pdf
Laursen,S.,Hassi,M.L.,Kogan,M.,Hunter,A.B.,&Weston,T.(2011).EvaluationoftheIBL
mathematicsproject:Studentandinstructoroutcomesofinquiry-basedlearningincollege
mathematics.UniversityofColoradoatBoulder.Retrievedfrom:
www.colorado.edu/eer/research/documents/IBLmathReportALL_050211.pdf
Laursen,S.L.,Hassi,M.L.,Kogan,M.,&Weston,T.J.(2014).Benefitsforwomenandmenofinquirybasedlearningincollegemathematics:Amulti-institutionstudy.JournalforResearchin
MathematicsEducation,45(4),405-418.
Martin,W.G.,&Gobstein,H.(2015).Generatinganetworkedimprovementcommunitytoimprove
secondarymathematicsteacherpreparation:Networkleadership,organization,andoperation.
JournalofTeacherEducation,66(5),482-493
Wilson,R.(1997).Adecadeofteachingreformcalculushasbeenadisaster,criticscharge.Chronicleof
HigherEducation,43(22),A12-A13.
Lawler,B.R.,Ronau,R.N.,&Mohr-Schroeder,M.J.(Eds.).(2016).ProceedingsofthefifthannualMathematics
TeacherEducationPartnershipconference.Washington,DC:AssociationofPublicLand-grantUniversities.