数学与数学教育者的对话市公开课一等奖百校联赛特等奖课件.pptx

數學與數學教育者對話林福來教授臺灣師範大學數學系第1页有這樣職業！HsingchivonBergmannAssociateProfessorDepartmentofdentistry,TheuniversityofBritishColumbiaMajorandresearchareasCurriculumandinstructionindentaleducation;large-scaleinternationalcomparativestudies;problem-basedlearning;inquiryteaching;collegescienceteachingandevaluationetc.第2页Shouldmathematiciansbere-educatedasmathematicseducator-researcher?(Lin,1988)YouhavedisciplineknowledgeandIhavemethodology.CompetenceinwritingacademicpapersResearch on MathematicsResearch on Mathematics Education第3页VisionMathematiciansandMathematicsEducator-ResearchersasCo-Learners第4页CooperationExampleAdevelopmentalProgramonChildrenMathematicsConceptsDevelopmentinTaiwan第5页TerminologyforMathematicsLearningNumeracy(England)CommonSense(Netherlands)Literacy(PISA)Competence(PISA)Proficiency(NCTM)瞭解與見解（臺灣）數學素養（臺灣）第6页數學素養研究走向研究對象教師學生教材內容第7页數學素養研究走向數學素養評量試題設計工作坊第8页數學素養研究走向國際性數學素養調查（如PISA,TIMSS)二階分析Chiu,M.-S.().Theinternal/externalframeofreferencemodel,big-fish-little-pondeffect,andcombinedmodelformathematicsandscience.Journal of Educational Psychology,104(1),87-107.Chiu,M.-S.().Achievementsandself-conceptsinacomparisonofmathandscience:Exploringtheinternal/externalframeofreferencemodelacross28countries.Educational Research and Evaluation,14(3),235-254.Chiu,M.-S.().Differentialpsychologicalprocessesunderlyingtheskill-developmentmodelandself-enhancementmodelacrossmathematicsandsciencein28countries.International Journal of Science and Mathematics Education,10,611-642.第9页數學素養教材發展與教學實驗DevelopingteachingmodulesDesigningassessmenttoolsExaminingstudentslearningorientation(e.g.,attitude,belief)第10页大眾數學素養調查案例大眾科學素養研究黃台珠、洪振方、周進洋、邱鴻麟、吳裕益、趙大衛（）。國民對科學與技術瞭解、興趣與關切度調查。行政院國家科學委員會計畫。第11页培養學生數學素養進路ExperiencingtheessenceofmathematicslearningOrigins of MathematicsWithin School MathematicsBeyond School MathematicsCommon sense第12页學習進路數學建模臆測閱讀了解探究教學概念診斷.第13页A Developmental Program on Children Mathematics Concepts Development in TaiwanFou-LaiLinDepartmentofMathematics,NationalTaiwanNormalUniversityConferenceonCommonSenseinMathematicsEducation:TheNetherlandsandTaiwan1923,Nov.14第14页AbstractThistalkwilldescribeafive-yearson-goingresearchprogramonchildrenmathematicsconceptsdevelopmentconductedinTaiwan.Morethanseventymathematicseducators,mathematicians,teachersandgraduatestudentsparticipatedinthisprogram.15第15页Yet,morethanahalfofthemareintheirfirstexperienceofbeinginvolvedinmathematicseducationresearch.Duringthisspecialbi-nationalconference,thisdevelopingprogramistakingachancetobeexaminedbythemethodologydevelopmentalresearchandseekforsuggestionsfromyou,thedevelopersofthismethodology.Mymainfocuswillbeonthelearningofthosenewresearchers.16第16页.The research program(08,07,)ConceptDevelopment:MathematicsinTaiwan(CD-MIT)第17页1.GoalsoftheCD-MITTodescribetheprocessandmechanismofstudentsmathematicsconceptdevelopmentinTaiwancompulsoryeducation.Toestablishandvalidateindigenouslearningandinstructionaltheoriesofmathematics.Toeducateresearchersinthefieldofmathematicseducation.18第18页192.TheProjects:MathematicsTopicsStudied第19页20第20页3.ChildrensInformalKnowledge(I.K.)Aperspectiveoflearning:InformalBridgingFormalFrameworkKnowledge21第21页22 INTENTIONAL AUTOMATICThe representation DENOTES the representationthe represented object in a:IS THE OUTCOME of a direct access to objectdiscursive situations nondiscursive situations non-aware memorized(reasoning operators)（visualization）implicit concept/theory facts images CognitiveArchitectureeg.Definingarectangle(longsquareinChinese)第22页4.FocusesonI.K.andConceptRepresentation(1)Apprehension/Recognizationeg:Apprehensionoffigure(Duval,1995)perceptual,sequential,manipulativeanddiscursiveapprehensionRecognizingapattern(Bishop,)Manipulative,proportional,recursive,functionalapproach.23第23页(2)ChildrensTheory-in-actioneg.Cognitivetheoryforpractice(Vergnaud,1998):concept-in-actionandtheory-in-actionRightangleHorizontalVertical24第24页(3)IntuitiveRuleseg.Thefourintuitiverules(Stavy&Tirosh,)MoreA-MoreB,SameA-SameB,Measurements(Chen,)Shapes(Chang,)25第25页(4)Visualizationeg.ComputerModels(ref:Tso,thisconference)(5)Representationeg.Function(Chang,)26第26页5.FocusesonCognitiveStrategies(1)(Informal)Reasoning/Inferenceseg.Exploringthedefinitionandpropositionsofgeometricshapes.(Lin,et.al,)(2)Symbolseg.Linearequation(Wu,)27第27页(3)Analogyeg.Definingarectanglevs.definingasquare.(Lin,et.al,)(4)Proportionalityeg.Recognizingapattern.(Lin,et.al,)(5)Modelingeg.Linearequation(Wu,)Infinity(Wang,)28第28页6.FocusesonSocial-CulturalCognition(1)Languageeg.Percentageofstudentswhorespondedthatasquareisarectangle(longsquareinChinese)Grader%29K1234567890335961 45 15732第29页(2)IndigenousChildreneg.CriteriausedtoclassifyshapesEllipseisassociatedwithrectanglebyaboriginechildren.(5/6)Ellipseisassociatedwithcirclebynon-aboriginechildren89%,63%,65%of7,8,9gradersrespectively30第30页(3)Superstitioneg.Red-envelopeandwhite-envelopevs.Evennumberandoddnumberpronunciationofthedigits4;6;8.Tenthousanddollarsiscalledonedollar(affective)Is1000anevenorodd?31第31页(4)CognitiveStyleseg.TheCramIndustryLearningbyexamples/ImitatingPracticingmakesyouskillful.32第32页 TopicsFocusesMathsArgumentationTimes Fraction FunctionInformalKnowledgeStrategiesSocial-cultural Context7.InfrastructureoftheResearchProgram33第33页.AResearcherasaLearner:theDevelopmentofCD-MIT第34页1.MondaysSeminar(1)SeminaronMathematicsEducationPublicationsLastedformorethantenyearsKluwersMathematicsEducationLibraryBooksfromFrendentalInstitute(OW&OC;CD-B)StudiesinMathematicsEducationSeries,TheFalmer.35第35页OthersHandbookofresearchonMathematicsTeachingandLearning.ThoughtandLanguage;L.Vygotsky,MITProblemsofrepresentationintheteachingandlearningofmathematics,LEASpeakingMathematically,RKPMathematicalExperience,andmanyothers.36第36页Now,theseminarfocusesonthebook:MathematicsEducationasaResearchDomain:Asearchforidentity.37第37页(2)Bi-WeeklyworkshopsontheprojectsbusinessClarifyingPresentingModifyingRe-designingMakingsenseoftheproject38第38页2.Two-daysWorkshopineachSemesterReportingtentativeresultsCommunicatingtheresearch39第39页3.RegularWeeklyProjectMeetingDesigningstudiesReviewingLiteratureAnalyzingdata40第40页4.MethodsUsedintheProgramCasebasedContextbasedComputerbasedClinicalInterviewPilotstudywithsomeclassesinlocalregionQuantitativestudy,anationalstudywithsampleofabout1600fromeachagepopulation.Teachingexperiment41第41页.ConceptualizingConceptDevelopment第42页1.ExerciseQuestionWhatdowemeanaboutconceptdevelopment?(questionraisedbyoneprojectdirectoron14,10,01)ExerciseWouldeveryoneexplainyourideasaboutconceptdevelopment?(05,11,01)43第43页Participants:Projectdirectors(17)Graduatedstudents(4)Teachers(2)44第44页2.Data15participantsareabletodescribetheirideasduringtheworkshop.Variousideasaboutconceptdevelopment45第45页BasicallytheyareatthecertaindegreeofVygoskyInformalvs.FormalSpontaneousvs.ScientificFromdailylifevs.Fromschool46T1,2,3(Ph.DStudents)(GroupDiscussion)第46页Concerningthechangesoftheconceptdevelopment：(1)Qualitativepresentation:abettercontrolofthecomplexityofconcepts(2)Strategymore systematizedwhen solving problems withconcepts(3)Quantitativepresentationthefacility of getting the rightanswers(4)Thepathoftheconceptdevelopmentisnotlinear,isrecursive.47T1,2,3(Ph.DStudents)(GroupDiscussion)第47页T4Presentingwithaviewofprocessconcept(1)Goal of study is to match themostappropriatetimeforlearningtheconceptwithstudentsgrowth.(2)Forexamples,statisticdiagrams:-Containingtheactivitiesofreadingthediagrams,gettinginformation to compose diagramsandexplainingdiagrams.48第48页Itsacomputerizingmodel.-Cultivating everyday experiencestogathersmallunitstogetherinmindasadatabase,viamentalorganization,and then expresstheconceptsinvariousways49T5:第49页T6:(1)The final destination of conceptdevelopmentistoknowhowtodefinetheconcept.(2)Thedevelopmentofconceptliketheprocess of cooperating by a midwifeand a sculptor.The former producesomethingandthelattergetridoftheimproper.(3)Beabletodistinguishexamplesandcounterexamplesunderdifferentcircumstances,thencanbeassessedinexpressingtheconceptsindifferentforms.(4)Andduringthedevelopmenttowardsthe goal,we need language andsymbols50第50页T7:(1)Itstheprocessofshuttlingbetweencertaintyanduncertainty.(2)ThesequenceofgrowthliketheprocessasTeacherdemo.(certain)Studentlearning(certain)Teachergivecounterexample(uncertain)Studentadjusted(certain)Teachergiveimproperexample(uncertain)Studentlearn(certain).51第51页T8:(1)Theintuitiveunderstandingisfromsemanticinterpretationofthewords.(2)Thenadjustedbytheconflictsandcounterexamplestowardsthefinalconcepts.(3)Forexample,independenteventsisintuitivelyviewedasdisjointevents52第52页T9:(1)Theprocessofgrowingislikeaconcentriccirclesmodel.(2)Itisageneticprocess.(3)Theacceptanceofdifferentdegreeofinaccuracyrevealsthelevelofgrowth.53第53页T10:Thecontentofknowledgeisformedbylotsofsubconcepts.For example,about linearfunction,elementary school children haveexperiencesofthecovarianceoftwovariables,butwithoutthewords;juniorhighschoolstudentsbegintolearntheterm,buttheymightviewy=f(x)=8isandy=8isnotalinearfunction,aftertheyadaptbothexamplesaslinearfunction;theythencometolearnthequadraticfunctionThroughtheprocessofinteriorizationandcondensation,thenabstracttothegeneralizedconcept.54第54页T11:Assumeoneisdeportedtoabarrenisland,onestartstoforgetoneusedtoknow,conceptisthelastbitofknowledgethatstillkeepinonesmind,itisnoteasytoforget.Thismetaphorcouldbeusedtobuildupthehierarchyamongconcepts.55第55页T12(1):UsetheconceptDevelopmentasanexampleofconcept.InterpretationtheconceptDevelopmentforexampleChickenisgrowingDuckisgrowingDogisgrowingChickenandduckbothareoviparous.Dogisviviparous.Theyarechangingfromsamll,hairlesstobigandwithhair.56第56页T12(2):Changesaretheessenceofconceptdevelopment-differentinvolume,differentinformsandgrowing.Changecanberevealedbyconceptmap,transition,association,evolutionanddegeneration.Developmentthencanbeusedtosaycitydevelopment.57第57页T13:(1)The inner characteristic of conceptshave to be emphasized and bounded.The character is more like an innerlanguage,notlanguageforcommunication.About concept,I still dont have itsdefinition.It can be explained by the envelopmodel(envelopofcurve,surface)58第58页T13:(2)AconceptisenvelopedbyFeaturesoftheconceptSituationsterminologySymbolicrepresentationExamplesUnderthesuitablecircumstances,thecorrectusage of the thinking can be reached.Symbols,specialtermsandplentyofexamplesareneeded.Thenitwillreachthecompletionofconcept59第59页T14:(1)Frominformaltoformal(2)Thesequenceofteachingmaterialwillaffectstudentsconceptdevelopment.(3)Thedevelopmentofconceptcouldbeinterpretedwiththeaspectofonedimensionalhierarchicallevels,butsometimesalsowiththeaspectofmulti-dimensionsmodel.60第60页T15:(1)(1)Developmentisaprocesstowardsthestatusthatoneisabletoexpresstheconceptinaspecific,accurateandeconomicway.(2)Theprocessusuallyiscarryingwithcertainmisconceptions.(3)Forexample,theconceptsimilarityislinkedwithlookslikelikephotocopyenlargementlikethesameapproximateakin,etc.61第61页T15:(2)Oneimportantfeatureofsimilarityisthedirectionalpositionofthefigures.Thefinalstageisonecandefinetheconceptofsimilarityoftwofiguresasthedistancebetweenanytwocorrespondingpointsofthetwofiguresareproportional.Analogyisusedprevalentlyinrecognizingtrianglesandquadrilateral.62第62页3.LevelofUnderstandingOnaconcretelevelevolutingfromanintuitivelevel.TowardsalocaltheorylevelAimingageneraltheorylevel63第63页4.ComingtoUnderstandingInter-ProjectbasedlearningReadingSensemakingCommunicatingPresentingOralandwritingValue-evaluating64第64页.Evolutionary,StratifiedandReflexive第65页1.EvolutionaryResearchersItemdevelopment66第66页2.StratifiedNavelevelConcretelevelLocallearningtheorylevel(Localinstructionaltheorylevel)Indigenouslearningtheorylevel(Indigenousinstructionaltheorylevel)67第67页3.ReflexiveResearchersdevelopmentTheorydevelopment68第68页.EducatingaMathematiciantobeaMathematicsEducator第69页1.Motives2.StrategyCanitbeachievedbydevelopmentalresearchapproach?3.Context70第70页數學素養評量設計第一場次工作坊北區輔導教授：林福來 曹博盛 楊凱琳 謝闓如 陳建誠第71页/2/1772背景我國我國PISA評量成績評量成績(國際排名國際排名)閱讀科學數學164123125？以數學素養(MathematicalLiteracy)為关键評量成績退步第72页/2/1773工作坊設計辦理北、竹、中、南、高、宜六區種子教師工作坊每區3場次，每場次3小時報名教師3場次全程參與要求每位學員設計一個素養題組第73页/2/1774工作坊進行方式：第一場次說明數學素養評量趨勢及PISA評量架構及意涵分組討論升學考題與學校考題 vs.素養試題教師試題設計經驗分享回家作業：教師自行決定試題情境與單元內容設計一個題組（約24小題）與各組輔導教授及學員討論(網路、電話)第74页/2/1775工作坊進行方式：第二場次組內學員報告並討論設計試題內容分群報告，修訂各自試題設計技術回家作業：修訂自我試題設計選擇一至二個班級施測，分析學生答題類型或答題選項百分比第75页/2/1776工作坊進行方式：第三場次組內學員報告並討論試題與施測結果分群報告，討論試題與施測結果評析升學考試與校內考試試題設計回家作業：完成試題修訂寄回定稿試題供編印第76页/2/1777進入主題先來素養一下第77页現實問題體驗：打折與加稅消費者購買東西時先打折，再加稅先加稅，再打折消費者付錢一樣（數學結果相同）對商家呢？對政府稅捐機關呢？第78页現實問題體驗：聯合壟斷1某島國只有兩家石油企业，分別是台大石油企业與中華石油企业，假設這兩家石油企业不暗中勾結聯合壟斷，而各行其市，則台大石油每年可賺2千億，中華石油可賺1千億。假設台大石油跟中華石油暗中勾結聯合壟斷，哄抬油價，每年可共同獲利7千億。問：若兩家石油企业暗中勾結，怎样公平瓜分這7千億？(題目來源：作者整合Nash,J.F.1951與ShapleyL.S.兩篇文章後杜撰本題情境，若與真實世界雷同，純屬巧合。)第79页現實問題體驗：聯合壟斷2答案1：均分最公平。兩家企业各得7/2千億。答案2：台大石油獲利能力是中華石油2倍，按照獲利能力分配才公平，所以台大石油應得14/3千億，中華石油應得7/3千億。答案3：兩家石油企业勾結壟斷之後共同多出來盈餘(邊際貢獻)是7214千億，所以兩家均分這多得4千億才公平。所以台大石油得到4千億中華石油得到3千億第80页現實問題體驗：聯合壟斷3答案4：在相對於數學模型外部真實世界，兩家石油企业可採共識分錢（当前美國公共花費問題，比如水費、電費、電話費、高速公路過路費，大都採用此方法。答案5：兩家企业也可採談判解決模式（談判結局必須滿足以下條件才不算輸，同樣是在數學模型內與數學模型外真實世界建模）。第81页現實問題體驗：聯合壟斷4數學正確不代表就是真能够解決現實生活上問題！第82页什麼是數學素養？了解與見解了解：對一件事情明白。見解：經批判、反思後進一步提出個人看法。不是知道多少數學，而是能夠拿知道數學處理資訊第83页什麼是數學素養？OECD派PISA定義數學素養：個人能在多樣不一样情境之下，將情境問題轉化成數學問題、使用數學及詮釋數學能力。這素養包含了數學推理及使用應用數學概念、程序、事實、工具來解釋、描述及預測現象。它協助個人瞭解數學在世界上所饰演角色，能夠進行有根據評斷，並且針對個體在生活中需求運用或者投入數學活動，以成為一個有積極、關懷、以及反思國民。第84页什麼是數學素養？美國派NRC定義數學素養概念瞭解(Conceptualunderstanding)程序流暢(Proceduralfluency)策略運用(Strategiccompetence)適當推理(Adaptivereasoning)建設性意向(Productivedisposition)第85页數學素養評量以PISA為例PISA數學素養評量目标追蹤並報告十五歲學生在靠近中等教育結束時數學素養水準。提供政策制订者、教育相關人員及研究人員有關學生數學素養水準跨時間成長訊息。第86页PISA評量數學素養三維度架構數學內容知識(Mathematicalcontentknowledge)情境問題解決三步驟及內蘊數學力(Mathematicalprocessesandtheunderlyingmathematicalcapabilities)情境與脈絡(Contexts)第87页數學內容知識(MathematicalcontentKnowledgeMathematicalcontentKnowledge)變化和關係(Changeandrelationships)空間和形(Spaceandshape)量(Quantity)不確定性（Uncertainty）從學生面對數學物件關係來思索涵蓋數學內容包含方程式、代數、坐標系、平面及立體幾何、測量、數與單位、比與百分比、估測、資料搜集和整理、取樣及樣本空間、機率第88页PISA評量數學素養三維度架構數學內容知識(Mathematicalcontentknowledge)情境問題解決三步驟及內蘊數學力(Mathematicalprocessesandtheunderlyingmathematicalcapabilities)情境與脈絡(Contexts)第89页情境問題解決三步驟將情境問題轉化成數學問題(Formulatingsituationsmathematically)使用數學概念、事實、步驟、和推理(Employingmathematicalconcepts,facts,proceduresandreasoning)詮釋、應用及評鑑數學結果(Interpreting,applying,andevaluatingmathematicaloutcomes)第90页PISA評量數學素養三維度架構數學內容知識(Mathematicalcontentknowledge)情境問題解決三步驟及內蘊數學力(Mathematicalprocessesandtheunderlyingmathematicalcapabilities)情境與脈絡(Contexts)第91页情境問題解決內蘊數學力1.情境與數學間溝通(Communication)2.問題數學化(Mathematising)3.使用及轉換表徵(Representation)4.推理和論述(Reasoningandargument)5.發展策略(Devisingstrategies)6.使用符號、形式及術語與運算(Usingsymbolic,formal,andtechnicallanguageandoperations)7.使用數學輔助工具(Usingmathematicaltools)第92页1.情境與數學間溝通溝通：主要強調對情境脈絡了解辨識出脈絡裡存在問題及挑戰。了解歷程包含閱讀、解碼、了解各種呈現方式資訊（如敘述、圖表、影像、及動畫）用來形成一個關於情境脈絡問題心智模式。進一步形成數學問題第93页2.問題數學化問題數學化：主要將真實情境脈絡問題轉化成一個數學形式。將情境結構化或概念化找出主要變因澄清與定義情境中假設、變數、關係、和限制。給出數學模式第94页3.使用及轉換表徵使用及轉換表徵：用數學表徵呈現真實情境脈絡。包含方程式、圖表、圖形、文字敘述、具體物，及各種表徵之間轉化。第95页4.推理和論述推理和論述：主要應用邏輯思索能力判斷以某種數學表徵呈現情境脈絡合理性。包含解釋、辯駁或檢證所形成數學表徵。第96页5.發展策略發展策略：發展及決定解決問題策略。辨別或發展或給出數學形式答案第97页6.使用符號、形式及術語與運算使用符號、形式