Review of Interactive Mathematics Program (IMP) at

ReviewoftheInteractiveMathematicsProgram(IMP)H.WuDepartmentofMathematics#3840UniversityofCalifornia,BerkeleyBerkeley,[e-mail]25,2000:Thisreviewhasbeencir-culatedsinceMarchof1992underthetitle,ReviewofCollegePreparatoryMathematics(CPM)atBerkeleyHighSchool.Initsoriginalform,itwasareportcommissionedinJanuaryof1992bytheBerkeleyUnifiedSchoolDistrictonanewcurriculum(calledCPMatthetime)beingintroducedinalocalhighschool.Fromthebeginning,however,IintendedthereporttobeareviewofthemathematicsofCPMwithonlyaminimalreferencetothelocalhighschool.Forthisreason,ithasbeenmetwithsomeinterestthroughtheyears.Theversionofthisreportbeingofferedbelow0Iamverymuchindebtedtomyfriendswhosehelpinvariousformsbackin1992madethewritingofthisreviewpossible:ProfessorsP.Chernoff,T.Y.Lam,B.Parlett,M.H.Protter,J.Sethian,andP.VojtaofUCBerkeley,Dr.IanBrownofLawrenceBerkeleyLaboratory,andProfessorsS.Y.ChengandR.E.GreeneofUCLA.IshouldalsorecordthelistofthepeoplewhomIhaveconsultedinthewritingofthisreviewandthankthemallfortheircourtesy,withoutimplyinginanywaythattheyareresponsibleforthefactualaccuracyofwhatfollows:Ms.LynneAlper,ProfessorDanFendel,Ms.SherryFraser,Ms.HeidiBoley,andMr.HarveyGarn.Finally,specialthanksarealsodueMr.ToddBoleyandMr.FredDunn-Ruizfortheircriticalcommentswhichledtosignificantimprovementsintheexposition.12datesbackessentiallytoJulyof1997,anditdiffersfromtheorig-inalonlyintheomissionofallreferencestoBerkeleyHighSchool,changingtheerstwhileCPMtoIMP(whichishowthiscurricu-lumnowchoosestoidentifyitself),clarifyingthemeaningoftheword“tracking”asusedintheoriginalversion,andaddingafewfootnotestobringsomeoftheoriginalcommentsup-to-date.Inearly1997,thefirstIMPtextfinallyappearedinprint(In-teractiveMathematicsProgram,Year1,KeyCurriculumPress,Berkeley,CA,1997),andbynowallfourvolumeshaveappeared.Otherthantheexpectedpedagogicalandexpositoryrefinements,thevirtuesanddefects—suchasIperceivedthemanddiscussedinthereport—ofthepreliminary1991versionmadeavailabletomebackin1992haveinthemainsurvivedinthepublishedver-sion.Inparticular,thereservationsagainstIMPdetailedin§§IIIandIVbelowregardingitslackofprecisionanditsinattentiontomathematicalclosureapplyequallywelltothe1997text.ThusIbelievethisreviewstillservesapurpose.MyrecommendationagainsttheuseofIMPforfuturecollegestudentsinscience,en-gineeringand(ofcourse)mathematicsisinmyviewasvalidnowasbefore.Ontheotherhand,twoadditionalcommentsmustbemade.ThefirstisthatonevirtueofIMPwhichIfailedtonoticein1992andwhichisbecomingincreasinglyapparenttomeisitsabilitytoputstudentsateaseinworkingonwordproblems,nomatterhowlong.TheabilitytoreadordinaryEnglishinthecontextofmathematicsis,paradoxically,aqualitysorelylack-ingamongcollegestudents.ItismuchtoberegrettedthatIMPcouldnotcapitalizeonthisachievementtolaunchstudentsintosubstantivemathematics.Theothercommentconcernswhether,eightyearsafterIwrotethisreview,hindsighthaschangedmyinitialjudgmentonthesuitabilityofIMPforusebystudentswhoeitherdonotgotocollegeorwillnotpursuescientificstudiesincollege.Havingreadquiteafewhighschooltextsintheinter-veningyearsofboththereformandthetraditionalvarieties,Ifinditdifficulttogiveasimpleanswer.Withoutadoubt,IMPismathematicallyflawedevenforthesestudents.Itisnotwith-outmerits,however(see§§III),andasofMarch2000,Iknowofnohighschoolmathematicstextbookseriesthatisclearlysupe-riorineveryway,andmanyaresubstantiallyworse.Withthe3newmathematicstextbookadoptioninCalifornialoominginthehorizon,thereisreasonforoptimismthatatleastsomealgebraItextswillbebetterintermsofprecision,rigor,andmathemati-calclosure.BecausethisadoptionhasnocontrolovertextbooksbeyondthealgebraIlevel,therewillbeabigareaofadvancedschoolmathematicsstillunaccountedfor.Withthisinmind,eachteachermustconsiderthetrade-offscarefullyintermsofhisorherneedsbeforemakingadecisionconcerningIMP.Mycon-servativerecommendationisthat,forstudentswhodonotgotocollegeordonotintendtopursuescientificstudiesincollege,allteacherswoulddowelltoconsultIMPoftenforsupplementarymaterialstobeusedintheclassroom.Contents§IPrologue[p.4]§IIIMP:anoverview[p.8]§IIIIMPasamathematicscurriculumforGroup1[p.9]§IVIMPasamathematicscurriculumforGroup2[p.15]§VSummary[p.25]§VIAppendix1:Whyisthequadraticformulaimpor-tant?[p.30]§VIIAppendix2:Miscellany[p.32]References4I.PrologueDuringhiscampaignforthepresidency(in1992),PresidentBushstatedthathewantedtoberememberedastheeducationpresidentand,inthesamebreath,thathewantedthehighschoolstudentsofthiscountrytobefirstinscienceandmathematics.Perhapswithoutrealizingit,hehadinfactsettwoseparategoalsforhispresidency:(1)thatallhighschoolgraduatesshouldbeliterateandscientificallyinformed,and(2)thatthehighschoolsshouldalsoproduceacorpsofpotentiallytopflightscientistsandengineers.Whileitisobviousthatthesetwogoalsinnowaycontradicteachother,itisnotusuallyrealizedthatthemethodsofachievingthemarenecessarilydistinct.Aneducationalprogramdesignedstrictlytoachievethefirstobjectivewouldmostlikelyproducestudentsthataretoodeficientinskillandknowledgetodoseriousscience,whileonedesignedexclusive