AP® Calculus and the Transition to

College Mathematics David Bressoud Macalester College, St. Paul, MN

T3Atlanta,GAMarch6,2010

PowerPointavailableatwww.macalester.edu/~bressoud/talks

MAA

The Chronicle of Higher Education January17,2010

TheRockyTransi/onFromHigh‐SchoolCalculus

hEp://chronicle.com/arHcle/High‐School‐Calculus‐The‐E/63533/

We need to:

Get more information about what happens to students who study calculus in high school.

Establish and enforce guidelines for high school programs offering calculus.

Re-examine first-year college mathematics.

AdvancedMathema-csandScienceCoursetakingintheSpringHighSchoolSeniorClassesof1982,1992,and2004.NCES2007‐312

⅓≥Precalculus¾≥AlgebraII

CBMSandCollegeBoarddata

2009:305

0

50

100

150

200

250

300

FallEnrollmentsinCalculusIversus

APCalculusExams(thousands)

4‐yearcolleges

2‐yearcolleges

APexams(AB&BC)

550,000–600,000studentsstudiedcalculusinhighschoolthisyear,roughly1/3ofthe1.8millionwhowillgodirectlyfromHStocollege.

2009:305

1999:158

1989:74

1979:25

Between150,000and200,000ofthesestudentsearnandtakeadvantageofcreditforCalculusI.

0

20

40

60

80

100

120

1980 1985 1990 1995 2000 2005

FallEnrollments,CalculusII(thousands)

4‐yearcolleges

2‐yearcolleges

CBMSdata

ItappearsthataverysignificantnumberofstudentstaketheirAPcreditanduseittoavoidtakinganycalculusincollege.

High school calculus should be pushing more students toward the study of higher mathematics in college.

It appears to be having exactly the opposite effect.

We need to:

Get more information about what happens to students who study calculus in high school.

Establish and enforce guidelines for high school programs offering calculus.

Re-examine first-year college mathematics.

Morgan&Klaric,2007:studyof22collegesanduniversiHesinfall,1994;gradesweightedsothatSATscoresarecomparable

Placedvia averagegradeinCalculusII SATAdjustedgrade

PassedCalculusI 2.43

3onABexam 2.69 2.64

4onABexam 2.90 2.78

5onABexam 3.34 3.15

Placedvia averagegradeinCalculusII SATAdjustedgrade

PassedCalculusI 2.50

3onBCexam 3.00 2.92

4onBCexam 3.45 3.35

5onBCexam 3.46 3.27

BarnardCollege,BinghamtonU.,BrighamYoungU.,CarnegieMellonU.,CollegeofWilliam&Mary,CornellU.,Dartmouth,GeorgeWashingtonU.,GeorgiaInsHtuteofTechnology,MiamiU.(Ohio),NorthCarolinaStateU.,TexasA&M,U.ofCaliforniaatDavis,U.ofIllinoisatUrbana/Champaign,U.ofIowa,U.ofMaryland,U.ofMiami,U.ofTexasatAusHn,U.ofVirginia,U.ofWashington,WesleyanCollege,WilliamsCollege

Keng&Dodd,2008:studyatUT‐AusHn,1998–2001;studentsselectedbystraHfiedrandomsamplesothatSATdistribuHonmatchedthatofAPstudents.

Prepara/onforCalculusII Averagegrade

a)3orhigheronBCexam 3.43

b)PassedCalculusI,SATmatchedtoBCstudents 3.16

c)3orhigheronABexam 3.13

d)PassedCalculusI,SATmatchedtoABstudents 3.03

e)3orhigheronABexamandtookCalculusI 2.96

f)Dualenrollmentcredit 2.93

g)BCcourse,nocredit,tookCalculusI 2.82

h)ABcourse,nocredit,tookCalculusI 2.45

StaHsHcallysignificantat0.05allfouryears:a)overb)twooffouryear:c)overd)oneoffouryear:c)overe)

TwoCurrentStudies:1.  PhilSadler,Harvard,FactorsInfluencingCollege

SuccessinMathema-csHSfactorsthatinfluencesuccessinCalculusI

2.  MAA(Bressoud,Carlson,Pearson,Rasmussen),Characteris-csofSuccessfulProgramsinCollegeCalculusCollegefactorsthatinfluencesuccessinCalculusI

andcasestudyanalysisofsuccessfulprograms.

From the transcript analysis of the National Education Longitudinal Study begun in 1988andCollegeBoardandCBMSdata.

Theanecdotalevidenceisstrongthattoday’sstudentswhodonotqualifyforcollegecredit(about350,000)struggletoarHculatehighschoolandcollegemathemaHcs.

Ofthehighschoolstudentswhograduatedin1992andstudied“calculus”whileinhighschool,31%tookprecalculusincollege,andafurther32%tooknocalculusincollege.

We need to:

Get more information about what happens to students who study calculus in high school.

Establish and enforce guidelines for high school programs offering calculus.

Re-examine first-year college mathematics.

W. L. Duren, Jr., University of Virginia, chair

E. G. Begle, Stanford University A. A. Blank, New York University Ralph P. Boas, Northwestern University Leslie A. Dwight, Southeastern State College Marion K. Fort, University of Georgia Samuel Goldberg, Oberlin College Edwin E. Moise, Harvard University Henry O. Pollak, Bell Telephone Laboratories G. Baley Price, University of Kansas A. W. Tucker, Princeton University R. J. Walker, Cornell University Gail S. Young, Jr., Tulane University

MathemaHcs1:IntroductoryCalculus1.  (7lessons)IntroducHon:extremevalueproblems,slopes,velocity

andrateofchange,limitsandapproximaHons2.  (14lessons)TechniquesandapplicaHonsofdifferenHaHon3.  (18lessons)TechniquesandapplicaHonsofintegraHon

MathemaHcs2(opHon1):1.  (9lessons)AdvancedunderstandingofdifferenHalandintegral

calculus2.  (15lessons)Limits:sequencesandseries,Newton’smethod3.  (6lessons)ParametricrepresentaHonofcurves,arclength4.  (9lessons)First‐orderdifferenHalequaHons

MathemaHcs2(opHon2):1.  (6lessons)Vectorsin3‐spce2.  (12lessons)DifferenHalcalculusoffuncHonsofseveralvariables3.  (9lessons)Advancedunderstandingofintegralcalculus4.  (12lessons)First‐orderdifferenHalequaHons

APCalculusAB

APCalculusBC

TryingtorestrictthegrowthofAPCalculusisanon‐starter:

• SuccessinAPCalculusisthesinglemostusefulpredictorofsuccessfulcompleHonofcollege.

• StudentswithoutaccesstoagoodprograminAPCalculusareatacompeHHvedisadvantage.

• ExxonMobilandtheGatesFoundaHonarestronglybehindthespreadofqualityprogramsinAPCalculus.

Severalstates,includingHawaii,Mississippi,SouthCarolina,andTennessee,havestatehighschoolstandardsthatdescribeawatereddownversionofcalculus.

California’sstandardsprescribeacoursethatisequivalenttoafullyearofcollegecalculus.

Nostatestandardstalkaboutthegoalsofthecourseorprescribepre‐requisiteknowledge.

We need to:

Get more information about what happens to students who study calculus in high school.

Establish and enforce guidelines for high school programs offering calculus.

Re-examine first-year college mathematics.

Number 1 complaint from high school teachers:

The focus of college placement tests is on what my students don’t know, rather than on what they know.

Colleges need to pay more attention to building on what students understand and can do.

2001–06 study at Arizona State University

Of students who took pre-calculus and • Their declared major required at least one semester of calculus, and • They earned an A in pre-calculus,

43% chose not to enroll in calculus.

During the period fall 2001 through fall 2006,

43% of engineering majors, 54% of mathematics majors,

51% of physical science majors, and 50% of technology majors

who enrolled in Calculus I at ASU and whose intended majors required Calculus II never earned credit for Calculus II.

“AfocusonteachingmustavoidthetemptaHontoconsideronlythesuperficialaspectsofteaching:theorganizaHon,tools,curriculum,content,andtextbooks.TheculturalacHvityofteaching–thewaysinwhichtheteacherandstudentsinteractaboutthesubject–canbemorepowerfulthanthecurriculummaterialsthatteachersuse.”

Stigler and Hiebert (2004), Improving mathematics teaching, Educational Leadership

USMilitaryAcademy,WestPoint:

MA103:MathemaHcalModelingandIntroducHontoCalculus,thefirstofourcorecoursesforallcadets.ThisisthefirstoffourcoursesintheUSMAmathemaHcscorecurriculum.ThefocusofthecourseistouseeffecHveproblemsolvingandmodelingtechniquestofindsoluHonstocomplexandotenill‐definedproblems.ThecourselaysthefoundaHonforcalculusanddifferenHalequaHonsthroughdifferenceequaHons.

MA104:CalculusI.ThiscoursebuildsuponthefoundaHonlaidinMA103,asthecadetlearnsaboutdifferenHalcalculusinsingleandmulH‐variableproblems.

This program has now been in place for twenty years.

CBMS data

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

1990 1995 2000 2005

per

cen

t o

f se

ctio

ns

year

Graphing Calculators

Research Univ Comprehensive Univ Undergrad college 2-year

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

1990 1995 2000 2005

per

cen

t o

f se

ctio

ns

year

Computer Assignments

Research Univ Comprehensive Univ Undergrad college 2-year

CBMS data

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

1990 1995 2000 2005

per

cen

t o

f se

ctio

ns

year

Writing Assignments

Research Univ Comprehensive Univ Undergrad college 2-year

CBMS data

0%

5%

10%

15%

20%

25%

30%

35%

1990 1995 2000 2005

per

cen

t o

f se

ctio

ns

year

Group Projects

Research Univ Comprehensive Univ Undergrad college 2-year

CBMS data

0%

2%

4%

6%

8%

10%

12%

Frac/onofincomingfreshmenintendingtomajorin...

Engineering

BioScience

PhysicalScience

ComputerScience

MathemaHcs

0%

2%

4%

6%

8%

10%

12%

Frac/onofgradua/ngseniorswhomajoredin...

Engineering

BioScience

PhysicalScience

ComputerScience

MathemaHcs

Source:TheAmericanFreshman

Source:NaHonalCenterforEducaHonStaHsHcs

CRAFTYCurriculumFounda-onsProject:

VoicesofthePartnerDisciplines

Biology: “Statistics, modeling and graphical representation should take priority over calculus.”

AmericanAssociaHonofMedicalCollegesandtheHowardHughesMedicalInsHtute,

Scien-ficFounda-onsforFuturePhysiciansSeptember,2009

Competency E1

Apply quantitative reasoning and appropriate mathematics to describe or explain phenomena in the natural world.

LearningObjecHves:

1.  DemonstratequanHtaHvenumeracyandfacilitywiththelanguageofmathemaHcs.

2.InterpretdatasetsandcommunicatethoseinterpretaHonsusingvisualandotherappropriatetools

3.MakestaHsHcalinferencesfromdatasets.

4.ExtractrelevantinformaHonfromlargedatasets.

5.MakeinferencesaboutnaturalphenomenausingmathemaHcalmodels.

6.Applyalgorithmicapproachesandprinciplesoflogic(includingthedisHncHonbetweencause/effectandassociaHon)toproblemsolving.

7.  QuanHfyandinterpretchangesindynamicalsystems.

“ThemathemaHcsprofessionasawholehasseriouslyunderesHmatedthedifficultyofteachingmathemaHcs.”

RameshGangolliMERWorkshopMay31,1991

PowerPointavailableatwww.macalester.edu/~bressoud/talks