数学与数学教育者的对话.ppt

數學與數學教育者的對話,林福來教授臺灣師範大學數學系,有這樣的職業！,Hsingchi von BergmannAssociate ProfessorDepartment of dentistry,The university of British ColumbiaMajor and research areasCurriculum and instruction in dental education;large-scale international comparative studies;problem-based learning;inquiry teaching;college science teaching and evaluation etc.,Should mathematicians be re-educated as mathematics educator-researcher?(Lin,1988)You have discipline knowledge and I have methodology.Competence in writing academic papers,Research on Mathematics,Research on Mathematics Education,Vision,Mathematicians and Mathematics Educator-Researchers as Co-Learners,Cooperation Example,A developmental Program on Children Mathematics Concepts Development in Taiwan,Terminology for Mathematics Learning,Numeracy(England)Common Sense(Netherlands)Literacy(PISA)Competence(PISA)Proficiency(NCTM)瞭解與見解（臺灣）數學素養（臺灣）,數學素養的研究走向,研究對象教師學生教材內容,數學素養的研究走向,數學素養的評量試題設計工作坊,數學素養的研究走向,國際性數學素養調查（如PISA,TIMSS)的二階分析Chiu,M.-S.(2012).The internal/external frame of reference model,big-fish-little-pond effect,and combined model for mathematics and science.Journal of Educational Psychology,104(1),87-107.Chiu,M.-S.(2008).Achievements and self-concepts in a comparison of math and science:Exploring the internal/external frame of reference model across 28 countries.Educational Research and Evaluation,14(3),235-254.Chiu,M.-S.(2012).Differential psychological processes underlying the skill-development model and self-enhancement model across mathematics and science in 28 countries.International Journal of Science and Mathematics Education,10,611-642.,數學素養的教材發展與教學實驗,Developing teaching modulesDesigning assessment toolsExamining students learning orientation(e.g.,attitude,belief),大眾的數學素養調查,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（2007）。國民對科學與技術的瞭解、興趣與關切度調查。行政院國家科學委員會計畫。,培養學生數學素養的進路,Experiencing the essence of mathematics learning,Origins of Mathematics,Within School Mathematics,Beyond School Mathematics,Common sense,學習進路,數學建模臆測閱讀理解探究教學概念診斷.,14,A Developmental Program on Children Mathematics Concepts Development in TaiwanFou-Lai LinDepartment of Mathematics,National Taiwan Normal University2001 Conference on Common Sense in Mathematics Education:The Netherlands and Taiwan 1923,Nov.2001.,15,Abstract,This talk will describe a five-years on-going research program on children mathematics concepts development conducted in Taiwan.More than seventy mathematics educators,mathematicians,teachers and graduate students participated in this program.,16,Yet,more than a half of them are in their first experience of being involved in mathematics education research.During this special bi-national conference,this developing program is taking a chance to be examined by the methodologydevelopmental research and seek for suggestions from you,the developers of this methodology.My main focus will be on the learning of those new researchers.,.The research program(08,200007,2005)Concept Development:Mathematics in Taiwan(CD-MIT),18,1.Goals of the CD-MIT,To describe the process and mechanism of students mathematics concept development in Taiwan compulsory education.To establish and validate indigenous learning and instructional theories of mathematics.To educate researchers in the field of mathematics education.,19,2.The Projects:Mathematics Topics Studied,20,21,3.Childrens Informal Knowledge(I.K.),A perspective of learning:Informal Bridging Formal Framework Knowledge,22,INTENTIONAL AUTOMATICThe representation DENOTES the representationthe represented object in a:IS THE OUTCOME of a direct access to objectdiscursive situations non-discursive situations non-aware memorized(reasoning operators)（visualization）implicit concept/theory facts images,Cognitive Architecture eg.Defining a rectangle(long square in Chinese),23,4.Focuses on I.K.and Concept Representation,(1)Apprehension/Recognization eg:Apprehension of figure(Duval,1995)perceptual,sequential,manipulative and discursive apprehensionRecognizing a pattern(Bishop,2000)Manipulative,proportional,recursive,functional approach.,24,(2)Childrens Theory-in-actioneg.Cognitive theory for practice(Vergnaud,1998):concept-in-action and theory-in-action Right angle Horizontal Vertical,25,(3)Intuitive Ruleseg.The four intuitive rules(Stavy&Tirosh,2000)More A-More B,Same A-Same B,Measurements(Chen,2001)Shapes(Chang,2001),26,(4)Visualizationeg.Computer Models(ref:Tso,2001,this conference)(5)Representationeg.Function(Chang,2001),27,5.Focuses on Cognitive Strategies,(Informal)Reasoning/Inferenceseg.Exploring the definition and propositions of geometric shapes.(Lin,et.al,2001)(2)Symbolseg.Linear equation(Wu,2001),28,(3)Analogyeg.Defining a rectangle vs.defining a square.(Lin,et.al,2001)(4)Proportionalityeg.Recognizing a pattern.(Lin,et.al,2001)(5)Modelingeg.Linear equation(Wu,2001)Infinity(Wang,2001),29,6.Focuses on Social-Cultural Cognition,Languageeg.Percentage of students who responded that a square is a rectangle(long square in Chinese)Grader%,30,(2)Indigenous Childreneg.Criteria used to classify shapesEllipse is associated with rectangle by aborigine children.(5/6)Ellipse is associated with circle by non-aborigine children 89%,63%,65%of 7,8,9 graders respectively,31,(3)Superstitioneg.Red-envelope and white-envelopevs.Even number and odd numberpronunciation of the digits 4;6;8.Ten thousand dollars is called one dollar(affective)Is 1000 an even or odd?,32,(4)Cognitive Styleseg.The Cram IndustryLearning by examples/ImitatingPracticing makes you skillful.,33,7.Infrastructure of the Research Program,.A Researcher as a Learner:the Development of CD-MIT,35,1.Mondays Seminar,Seminar on Mathematics Education PublicationsLasted for more than ten yearsKluwers Mathematics Education LibraryBooks from Frendental Institute(OW CD-B)Studies in Mathematics Education Series,The Falmer.,36,OthersHandbook of research on Mathematics Teaching and Learning.Thought and Language;L.Vygotsky,MITProblems of representation in the teaching and learning of mathematics,LEASpeaking Mathematically,RKPMathematical Experience,and many others.,37,Now,the seminar focuses on the book:Mathematics Education as a Research Domain:A search for identity.,38,(2)Bi-Weekly workshops on the projects businessClarifyingPresentingModifyingRe-designingMaking sense of the project,39,2.Two-days Workshop in each Semester,Reporting tentative resultsCommunicating the research,40,3.Regular Weekly Project Meeting,Designing studiesReviewing LiteratureAnalyzing data,41,4.Methods Used in the Program,Case basedContext basedComputer basedClinical InterviewPilot study with some classes in local regionQuantitative study,a national study with sample of about 1600 from each age population.Teaching experiment,.Conceptualizing Concept Development,43,1.Exercise,Question What do we mean about concept development?(question raised by one project director on 14,10,01)Exercise Would everyone explain your ideas about concept development?(05,11,01),44,Participants:Project directors(17)Graduated students(4)Teachers(2),45,2.Data,15 participants are able to describe their ideas during the workshop.Various ideas about concept development,46,Basically they are at the certain degree of Vygosky Informal vs.Formal Spontaneous vs.Scientific From daily life vs.From school,T1,2,3(Ph.D Students)(GroupDiscussion),47,Concerning the changes of the concept development：(1)Qualitative presentation:a better control of the complexity of concepts(2)Strategymore systematized when solving problems with concepts(3)Quantitative presentationthe facility of getting the right answers(4)The path of the concept development is not linear,is recursive.,T1,2,3(Ph.D Students)(GroupDiscussion),48,T4,Presenting with a view of process concept(1)Goal of study is to match the most appropriate time for learning the concept with students growth.(2)For examples,statistic diagrams:-Containing the activities of reading the diagrams,getting information to compose diagrams and explaining diagrams.,49,Its a computerizing model.-Cultivating everyday experiences to gather small units together in mind as a database,via mental organization,and then express the concepts in various ways,T5:,50,T6:,(1)The final destination of concept development is to know how to define the concept.(2)The development of concept like the process of cooperating by a midwife and a sculptor.The former produce something and the latter get rid of the improper.(3)Be able to distinguish examples and counterexamples under different circumstances,then can be assessed in expressing the concepts in different forms.(4)And during the development towards the goal,we need language and symbols,51,T7:,(1)Its the process of shuttling between certainty and uncertainty.(2)The sequence of growth like the process as Teacher demo.(certain)Student learning(certain)Teacher give counter example(uncertain)Student adjusted(certain)Teacher give improper example(uncertain)Student learn(certain).,52,T8:,(1)The intuitive understanding is from semantic interpretation of the words.(2)Then adjusted by the conflicts and counterexamples towards the final concepts.(3)For example,independent events is intuitively viewed as disjoint events,53,T9:,(1)The process of growing is like a concentric circles model.(2)It is a genetic process.(3)The acceptance of different degree of inaccuracy reveals the level of growth.,54,T10:,The content of knowledge is formed by lots of subconcepts.For example,about linear function,elementary school children have experiences of the covariance of two variables,but without the words;junior high school students begin to learn the term,but they might view y=f(x)=8 is and y=8 is not a linear function,after they adapt both examples as linear function;they then come to learn the quadratic function Through the process of interiorization and condensation,then abstract to the generalized concept.,55,T11:,Assume one is deported to a barren island,one starts to forget one used to know,concept is the last bit of knowledge that still keep in ones mind,it is not easy to forget.This metaphor could be used to build up the hierarchy among concepts.,56,T12(1):,Use the concept Development as an example of concept.Interpretation the concept Development for exampleChicken is growingDuck is growingDog is growingChicken and duck both are oviparous.Dog is viviparous.They are changing from samll,hairless to big and with hair.,57,T12(2):,Changes are the essence of concept development-different in volume,different in forms and growing.Change can be revealed by concept map,transition,association,evolution and degeneration.Development then can be used to say city development.,58,T13:(1),The inner characteristic of concepts have to be emphasized and bounded.The character is more like an inner language,not language for communication.About concept,I still dont have its definition.It can be explained by the envelop model(envelop of curve,surface),59,T13:(2),A concept is enveloped by Features of the conceptSituationsterminologySymbolic representationExamplesUnder the suitable circumstances,the correct usage of the thinking can be reached.Symbols,special terms and plenty of examples are needed.Then it will reach the completion of concept,60,T14:,(1)From informal to formal(2)The sequence of teaching material will affect students concept development.(3)The development of concept could be interpreted with the aspect of one dimensional hierarchical levels,but sometimes also with the aspect of multi-dimensions model.,61,T15:(1),(1)Development is a process towards the status that one is able to express the concept in a specific,accurate and economic way.(2)The process usually is carrying with certain misconceptions.(3)For example,the concept similarity is linked with looks like like photo copy enlargement like the same approximateakin,etc.,62,T15:(2),One important feature of similarity is the directional position of the figures.The final stage is one can define the concept of similarity of two figures as the distance between any two corresponding points of the two figures are proportional.Analogy is used prevalently in recognizing triangles and quadrilateral.,63,3.Level of Understanding,On a concrete level evoluting from an intuitive level.Towards a local theory levelAiming a general theory level,64,4.Coming to Understanding,Inter-Project based learningReadingSense makingCommunicatingPresentingOral and writingValue-evaluating,.Evolutionary,Stratified and Reflexive,66,1.Evolutionary,ResearchersItem development,67,2.Stratified,Nave levelConcrete levelLocal learning theory level(Local instructional theory level)Indigenous learning theory level(Indigenous instructional theory level),68,3.Reflexive,Researchers developmentTheory development,.Educating a Mathematician to be a Mathematics Educator,70,1.Motives2.StrategyCan it be achieved by developmental research approach?3.Context,數學素養的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,2011/2/17,72,背景,以數學素養(Mathematical Literacy)為核心的評量成績退步,2011/2/17,73,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,2011/2/17,74,工作坊進行方式：第一場次,說明數學素養評量的趨勢及PISA的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的經驗分享回家作業：教師自行決定試題的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),2011/2/17,75,工作坊進行方式：第二場次,組內學員報告並討論設計的試題內容分群報告，修訂各自的試題設計技術回家作業：修訂自我的試題設計選擇一至二個班級施測，分析學生的答題類型或答題選項百分比,2011/2/17,76,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的試題設計回家作業：完成試題的修訂寄回定稿的試題供編印,2011/2/17,77,進入主題先來素養一下,現實問題的體驗：打折與加稅,消費者購買東西時先打折，再加稅先加稅，再打折消費者付的錢一樣（數學結果相同）對商家呢？對政府稅捐機關呢？,現實問題的體驗：聯合壟斷1,某島國只有兩家石油公司，分別是台大石油公司與中華石油公司，假設這兩家石油公司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：若兩家石油公司暗中勾結，如何公平的瓜分這 7 千億？(題目來源：作者整合與 Shapley L.S.兩篇文章後杜撰本題情境，若與真實世界雷同，純屬巧合。),現實問題的體驗：聯合壟斷2,答案 1：均分最公平。兩家公司各得 7/2 千億。答案 2：台大石油的獲利能力是中華石油的 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 3：兩家石油公司勾結壟斷之後共同多出來的盈餘(邊際貢獻)是 7214 千億，所以兩家均分這多得的 4 千億才公平。因此台大石油得到 4 千億 中華石油得到 3 千億,現實問題的體驗：聯合壟斷3,答案 4：在相對於數學模型外部的真實世界，兩家石油公司可採共識分錢（目前美國的公共花費問題，例如水費、電費、電話費、高速公路過路費，大都 採用此方法。答案 5：兩家公司也可採談判解決的模式（談判結局必須滿足下列條件才不算輸，同樣是在數學模型內與數學模型外的真實世界建模）。,現實問題的體驗：聯合壟斷4,數學正確不代表就是真的可以解決現實生活上的問題！,什麼是數學素養？了解與見解,了解：對一件事情的明白。見解：經批判、反思後進一步提出個人的看法。不是知道多少數學，而是能夠拿知道的數學處理資訊,什麼是數學素養？OECD派,PISA定義的數學素養：個人能在多樣不同的情境之下，將情境問題轉化成數學問題、使用數學及詮釋數學的能力。這素養包括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扮演的角色，能夠進行有根據的評斷，並且針對個體在生活中的需求運用或者投入數學活動，以成為一個有積極的、關懷的、以及反思的國民。,什麼是數學素養？美國派,NRC定義的數學素養概念的瞭解(Conceptual understanding)程序的流暢(Procedural fluency)策略的運用(Strategic competence)適當的推理(Adaptive reasoning)建設性的意向(Productive disposition),數學素養的評量以PISA為例,PISA數學素養評量的目的追蹤並報告十五歲學生在接近中等教育結束時的數學素養水準。提供政策制定者、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的訊息。,PISA評量數學素養的三維度架構,數學內容知識(Mathematical content knowledge)情境問題解決三步驟及內蘊的數學力(Mathematical processes and the underlying mathematical capabilities)情境與脈絡(Contexts),數學內容知識(Mathematical content Knowledge),變化和關係(Change and relationships)空間和形(Space and shape)量(Quantity)不確定性（Uncertainty）從學生面對的數學物件關係來思考涵蓋的數學內容包括方程式、代數、坐標系、平面及立體幾何、測量、數與單位、比與比例、估測、資料收集和整理、取樣及樣本空間、機率,PISA評量數學素養的三維度架構,數學內容知識(Mathematical content knowledge)情境問題解決三步驟及內蘊的數學力(Mathematical