ap calculus and the transition to college mathematicsbressoud/talks/2010/t3-transition.pdf · ap®...
TRANSCRIPT
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