Title Geometric Algebra New Foundations, New Insights标题的几何代数的新基础新的见解.docx

TheworkofAmbjornNaevewithinthefieldofMathematicsEducationalReformAmbjornNaeveMarch2001(I)CIDCentreforUserOrientedITDesignTableofContentTableofContent21. ProblemStatement42. Background52.1. TheCVAPresearchgroup62.2. TheGeometricToolboxproject62.3. PrqjectiveDrawingBoard:dynamicgeometricexplorations72.4. TheCentreforuser-orientedITDesign(CID)92.5. TheGardenofKnowledgeproject102.6. SomeILEprojectsatCID113. ConceptBrowsing123.1. Conceptualtopologies123.1.1. Traditionalconceptualtopologies123.1.2. Dynamicconceptualtopologies-hyperlinkedinformationsystems123.1.3. Problemswiththeabovementionedconceptualtopologies133.2. BasicDesignPrinciplesforConceptBrowsers143.3. Conzilla144. KnowledgeManifolds164.1. CreatingMultiplyNarratedKnowledgeComponents164.2. ComposingKnowledgeComponentsintoLearningModules175. MathematicalILEworkatCID195.1. TheVirtualMathematicalExploratorium195.1.1. Introduction195.1.2. Surfingthecontext215.1.3. Displayingmeta-datadescriptions235.1.4. Viewingthelistofcontentcomponentsthroughdifferentfilters245.1.5. Viewingtheactualcontentofacomponent275.1.6. Displayingthedecriptionofthecontentcomponents285.1.7. DesigngoalsandeducationalapplicationsoftheVME295.2. LiveGraphics3D-makingMathematicagraphicscomealive315.3. MathematicalComponentArchives325.3.1. GeneralgoalsoftheMCAproject335.4. CyberMath336. TheSwedishLearningLab376.1. Excerptfromtheproposalsummary376.2. NewMeetingPlacesforLearning386.2.1. Summary386.2.2. Projectoverview387. Archives,Portfoliosand3DEnvironments(APE)397.1. APE:TrackA:ContentandContextofMathematicsinEngineeringEducation397.1.1. Goals397.1.2. ActivityplanforstudywithintheMScprograminITatKTH397.1.3. ExcerptsfromtheProgressReportyear2000407.1.4. Currentstateoftheprojectcomparedtotheactivityplan417.1.5. Implementationofinteractivecontentandappropriatetools437.1.6. Educationalevaluation/assessmentresults(Study1)437.2. APE:TrackC:3DCommunicationandVisualizationEnvironmentsforLearning(CVEL)447.2.1. GoalsoftheCVEL-Project447.2.2. Activityplanyear2000(Workpackages)447.2.3. ExcerptsfromtheProgressReportyear20(X)-ResultsatDIS467.2.4. ExcerptsfromtheProgressReportyear2000-ResultsatCID468. PlannedLearningLabProjects:PersonalizedAccesstoDistributedLearningResources528.1. ProposalSummary528.2. Module:InfrastructureandIntelligentServices538.2.1. Contributors538.2.3. ProblemDescription,ResearchAspects538.2.4. ResearchGoalsZDeliverables538.2.5. Interactionwithothermodules558.2.6. CIDteam558.3. Module:Personalizedaccesstoeducationalmedia558.3.1. Contributors558.3.2. ResearchIssues569. GeometricAlgebraandMathEducationalReform589.3. DavidHestenes589.4. Modelingprogramexcellenceaward589.4.1. Exemplary&promisingeducationaltechnologyprograms(2000)599.5. SIGGRAPH-2000:GeometricAlgebraCourse609.5.1. CourseTitle:GeometricAlgebra:NewFoundations,NewInsights609.6. MathXplor-aVirtualMathematicsExploraioriumgroup629.7. DavidHestenesatKTH659.8. EducationalReformbasedonSoftwaredevelopment659.8.1. TheSimCalcProject659.8.2. OverallStrategy6710. CILTandEdGrid6710.3. TheCILT2000conferenceandtheM&Vworkshop6711. References6911.3. SelectedpublicationsbyAmbjomNaeve6911.4. SupervisoryActivitiesbyAmbjornNaeve701. ProblemStatementDuringthelasttwodecades,thespectacularadvancementswithinthefieldofinformationtechnologyhavecreatedpowerfulgraphicalworkstationswithpossibilitiestostudymathematicsinnewandexitingways.Todayitisevidenthowcomputerbasedanimationsandsimulationshaveaffectedmostfieldsthatinvolvemathematicalapplicationsinsomeway.Visualizationofstructuralrelationshipsanddynamicprocesseshasemergedasafieldofitsown,withapplicationswithinscienceandtechnologyaswellaswithineconomicsandsocialsciences.Itisthereforesomethingofaparadoxthatoneofthefieldsthatseemstohavebeenleastaffectedbythisdevelopmentisthefieldofmathematicsitself.Thisisespeciallytrueofmathematicseducation,i.e.mathematicaldidactics.Atouruniversitieswearestillcarryingonthetraditionalwaysofmathematicsteaching-executingourcoursesintheoverallspiritofuthesameprocedureaslastyear11!Amongmathematicsteachers,computersareoftenconsideredathreateningelement,thatfocusesthestudents'attentioninthewrongdirection.Thisattitudeiseasilyreinforcedbythemultitudeoflow-qualityeducationalsoftwarethatsupportsmathematicallytrivialpursuitsinonewayoranother.Infact,thisdeplorablestateofaffairshascausedtheissueofcomputer-supportedorcomputer-disturbedmathematicstosurfaceasamajordiscussionthemeamongmathematicaleducationalists. See the report titled Datorstddd eller (Iatorstord mutenutikundervisning ?, Hogskoleverkets Skriftserie I999:4S, ISSN 1400-9498.Ofcourse,therearemanyexceptionsfromthisbasicpatternintheformofindividualteachersthatinvolvethemselvesintryingtomakeintelligentuseofthepossibilitiesofpedagogicalrenewalthatareofferedbytheemergingICTtechnology.Butaslongastheyactalone,asisolatedenthusiastsemergedinanoceanofskeptics,theireffortsandexperienceswillremainhiddenandhencebedifficulttoharnessandreuseinasystematicway.2. BackgroundDuringmorethan3decadesAmbjornNaevehasbeeninvolvedinmathematicseducationatKTH-bothasateacherintheMathematicsdepartment,andasaneducationalreformist.Overthepast15years,Naevehasinitiatedandcoordinatedanumberofprojectsaimingtomakeuseofcomputersinordertoincreasethecomprehensibilityandaccessibilityofmathematicalconceptsandstructuresatalllevelsoftheeducationalsystem.Theseprojectshaveallbeenbasedonhisfirmconvictionthatincreasingthestudents1intuitiveunderstandingofmathematicalstructures-bothattheuniversitylevelandatthemoreelementaryschoollevel-isakeyelementinmotivatingthemtopursuemathematicalstudiesingeneral.ThisworkoriginatedasapartofNaeve,sresearchinthefieldofgeometricmodelingwithintheComputerVisionandActivePerception(CVAP)researchgroupatNADA.Duringthelast4yearsithasbeenapartofNaeve,sresearchworkinthefieldofinteractivelearningenvironmentsattheCentreforuser-orientedITDesign(CID)atNADA.Theworkhasresultedinanumberofsoftwaretoolsfortheinteractiveexplorationofmathematics12,severalofwhichhaveattractedbothnationalandinternationalattention.SomeofthemorerecentlydevelopedonesarePDB(ProjectiveDrawingBoard),COnZ川a(:Cid.nada.kth.se/il/ConZilla/default.html)andCyberMath(:nada.kth.se/qustavt/cvbermath/).TheywerepresentedlastyearattheSiggraphconferenceinNewOrleans-probablythemostprestigiouscomputerconferenceintheworld.CyberMathhasalsobeenacceptedforpresentationatICDE-2001(the20:thWorldConferenceonOpenLearningandDistanceEducation)inDusseldorfinApril(:icde.org).ConzillaandCyberMathhavebeenpartlydevelopedwithintheAPE(Archives,Portfolios,Environments)projectincooperationbetweenCIDandtheSwedishLearningLab(:/IeaminaIab.kth.se/library/Dresentations).TheprojectshaveinvolvedsystemsdesignandprogrammingbyNaevehimself,buthehasalsoheadedanumberofdevelopmentteamsinvolving3rdand4thyearstudentsattheComputerSciencedepartment(NADA)aswellasprojectsforthemastersthesis19,21,23andforthedoctoralthesis20.Thecommonthemeofalltheseprojectshasbeenthepresentationofmathematicalideasandstructuresinawaythatfacilitatesanincreasedunderstandingofthembymakingitpossibletoexploretheminteractivelythroughvariousformsofexperimentationandvisualization.Initially,theprojectswerefocusedongeometry-whichresultedinsoftwareprogramslikeMapCon(1986),MacWaIIpaper(1987)andMacDrawboard(1988),butprogramslikeHyperFIow(1987)andMapAnaIyze(1989)whichdealtwiththegeneralconceptoffunctionandPrimeTime(1992),whichdealtwithelementaryarithmeticwerealsocreated.See12forfurtherdetailsonthesesoftwaretools.Duringrecentyearstheworkhasbecomedirectedtowardsthecreationofcomputer-supportedmathematicstoolsthatcanfunctiontogetherinamodularizedanddistributedinteractivelearningenvironment.Inthisrespect,thefollowingprojectsdeservetobementioned: GeometricToolbox(researchworkatCVAP,1986-1994). TheGardenofKnowledge(interdisciplinaryprojectatCIDsince1996). ProjectiveDrawingBoard(doctoralworkatCVAP,1995-1999). Conceptualbrowsing(startedasamastersthesisworkatCID1998-1999,presentlyadoctoralthesisprojectatCID).Below,someoftheseprojectswillbedescribedbriefly.Foramoredetaileddescriptionthereaderisreferredto12.1.1. TheCVAPresearchgroupTheComputerVisionandActivePerception(CVAP)groupisaninternationallyrenownedresearchgroupatNADAwithinthefieldofComputerVisionandRobotics,whichhasbeendevelopedduringthepast15yearsundertheleadershipofProf.Jan-OlofEklundh.TheCVAPgrouphasastrongfoundationinmathematicsandcomputerscience.Ithasacquiredaninternationalreputationforitsabilitytotransformmathematicalideasintotechnicallyviableapplications,whichhasmadethegroupabletoattractalargenumberofgiftedstudents.Overtheyears,severalofthesocalled"excellencepositions"fordoctoralstudentsatKTHhavebeenheldbyresearchstudentsatCVAP.IncombinationwiththeinspiringleadershipofProf.Eklundh,thishasmadepossiblesomemathematicalsoftwaredesignofthehighestinternationalqualitylikee.g.GeometricToolbox(Naeve&Appelgren1986-94)andProjectiveDrawingBoard(Naeve&Winroth1995-1999).1.2. TheGeometricToolboxprojectThetypicalgeometricmodelingsituationoftodayischaracterizedby-andquitefrequentlyplaguedby-anumberoftoolswithahighdegreeofspecial"streamlined"performance.Thishasalmostinvariablyledto"adhoc'1choicesandsimplificationsthathavecreatedmathematicalinconsistenciesandtherebyrenderedalmostallofthetoolsincompatiblewiththeothers-preventingthemtoworktogetherinacoherentfashionagainstthesame',allinclusive"universalgeometricbackground.AtCVAP,AmbjornNaeveandJohanAppelgrenhavedevelopedasoftwarepackagecalledReflections19,whichisasystemfortheinteractivestudyofsurfaceshape.ThissystemwasusedasanexperimentalplatformforthetheorydevelopedbyNaeveinhisdissertation7.ReflectionsispartofasoftwaresystemcalledSurface-Geometry,whichisamathematicallybased,computationallyefficientgeometricrepresentationschemefor3Dsurface12.TheSurface-GeometrysystemisitselfpartofalargergeometricmodelingprojectwithinCVAP,calledGeometricToolbox,whichisaimedatproducinganinteractive"mathematics-friendly'geometricexperimentationenvironment-akindofgeometric"objectlibrary"-consistingofacollectionofcompatibleandreusablegeometricstructuresandalgorithmiccomponents.Usingthiskindofgeometrictoolbox,differentkindsofgeometricexperiments-putergraphics,computationalgeometryandcomputervision-canbeeasily"wiredtogether"andalltherelevantparameterscanbemanipulatedinamathematicallycontrolledandinteractivelyobservableway.Thedesiretoperformsuchexperiments-whereoneiscombining"heavycomputing"with"immediateviewing"oftheresult-isgrowingrapidlywithinthecommunityofcomputationalgeometry-asthepowerofsuchtechniquesindevelopingandtestingdifferentalgorithmsisbecomingmoreandmoreapparent.Thisisduetoacombinationoftheenormousincreaseofcomputationalpowerthathasmanifesteditselfinhardwarecomponentsoverthelastfewyearsandtheadvancedgraphicsworkstationcapabilitiesthatareonthevergeofsettlingdownoneverybody'sdesktop.Ithasfinallybecomefeasibletosimulatealargeclassofcomplicatedgeometricalsituationsandobtaininformationonlinewithdirectrelevancetotheunderstandingoftheunderlyingproblem.Thepossibilitytointeractivelyexpandonesintuitionaboutaproblem-byperformingmathematicallycontrolledexperimentsinthisway-isaverypowerfultechniquethatisboundtohaveaprofoundeffectontheentireresearchmethodologyofthefuture.ForadetaileddescriptionoftheresultsoftheGeometricToolboxproject,thereaderisreferredto6,7,12and19.1.3. ProjectiveDrawingBoard:dynamicgeometricexplorationsPDB(ProjectiveDrawingBoard)isaprogramthatsupportsinteractiveexplorationofgeometricconstructionsintheprojectiveplane PDB has been created by Harald Winroth as a part of his doctoral project 20 at CVAP under the supen,ision of Ambjorn Naeve. The program is based on an earlier prototype called MacDrawboard, which was developed by Ambjdm Naeve and a group of Computer Science students in 1998 12.Theprojectiveplaneisanenlargementoftheordinary(Euclidean)planewhichisconstructedbyintroducingnewelements(asetofidealpointsandoneidealline)insuchawaythattwoparallellinesintersectinanidealpointandalltheidealpointslieontheidealline.Intheprojectiveplane,twolinesalwaysintersectin(=lieon)onepoint,andofcoursetwopointsstilllieon(=intersectin)oneline.Hencethestructuralrelationshipsbetweenpointsandlinesbecomemuchsimpler,sincetheyarenowdevoidoftheclassicaleuclideanexceptionscausedbyparallellines,thatcanleadtocomplicatedcombinatorialproblems.Intheprojectiveplane,pointsandlinesareinfactrepresentedbythesamealgebra,anditisonlytheinterpretationofthealgebraicformulasthatdeterminetheirgraphicalappearence(asapointorasaline).Everygeometricconstructionhasahistory,whichreflectstheorderinwhichtheconstructionhasbeenbuiltup.Aconstructionprocesscanberegardedasaninterplaybetweenrandomchoice(e.g.choosetwopointsPandQ)andcanonicalnecessity(e.g.drawthelinePO).Ageometricobjectcanpartakeofboththeseelements(e.g.choosealineonP).Toanygeometricobjectwecanthereforeassociateasetofchildrenandasetofparentsinanaturalway.Intheexampleabove,thelinePOisachildofboththepointPandthepointQ,andbothofthesepointsareparentsofthelinePO.OneofthebasicideasinPDBistokeeptrackofthehistoryofageometricconstructionandmakeitpossibletochangeitinaconsistentway.Thismeansthatachangethataffectsanobjectacertainstageintheconstructionshouldpropagateforwardsothatitaffectsallthechildrenofthisobject.Inordertoupdatetheconstructioninthisway,PDBhasaccesstoanentirehierarchyofcoordinatesystemsthatkeeptrackofthepositionofeachobjectrelativetoitsparents.PresentlyPDBworksonlywiththeelementsofclassicalprojectivegeometry,i.e.points,linesandconics.However,thesystemisdesignedinaccordancewiththeobject-orientedparadigmanditiswellmodularizedtomakeiteasytoenlargeandexpandinvariousways.PDBpresentsbothagraphicandalogicviewofageometricconstruction.Moreover,theprogramallowsyounotonlytochangethepositionofanobject,withcoherentupdatesoftheeffectsonitschildren,buttoactuallychangethelogicoftheconstructionbydynamicallychangingtheconstraintsofanobject.Forexample,apointwhichisachildofaconiccanbetornofftheconicandeitherbeturnedintoafreepoint,orbesubjectedtosomeotherconstraintande.g.becomeapointonasomeline.Thisallowsyoutoplaywithaconstructionandexperiencepreciselyunderwhatconditionscertainthingshappen,i.e.youcaninteractivelyexploretheif-and-only-ifconditionsofageometrictheorem.GraphicviewLogicviewFig.1:Desargue,stheorem.Thecornersoftwotriangles(abcanda,b,c,)areperspectivefromapoint(green)ifandonlyifthecorrespondinglines(ab&a,b,be&b,c,andca&c,a,)ofthetrianglesareperspectivefromaline(black),i.e.thepointsp,q,rarecollinear.ToconveyanideaofthedynamicpossibilitiesofPDB,aQuickTimemoviethatillustratesthedynamicexplorationofDesargue1Stheoremisavailableat:nada.kth.se/amb/SnapzPro/Desargues.mov.1.4. TheCentreforuser-orientedITDesign(CID)CID,whichwasstartedin1995,isacompetencecenteratthedepartmentofNumericalAna