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數學與數學教育者的對話

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 Bergmann Associate Professor Department of dentistry, The university of British Columbia Major and research areas

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數學與數學教育者的對話

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  1. 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

  2. 有這樣的職業！ • Hsingchi von Bergmann • Associate Professor • Department of dentistry, The university of British Columbia • Major and research areas • Curriculum and instruction in dental education; large-scale international comparative studies; problem-based learning; inquiry teaching; college science teaching and evaluation etc.

  3. 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 Education Research on Mathematics

  4. Vision • Mathematicians and Mathematics Educator-Researchers as Co-Learners

  5. Cooperation Example • A developmental Program on Children Mathematics Concepts Development in Taiwan

  6. 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）

  7. 數學素養的研究走向 • 研究對象 • 教師 • 學生 • 教材內容

  8. 數學素養的研究走向 • 數學素養的評量試題設計工作坊

  9. 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.

  10. 數學素養的教材發展與教學實驗 • Developing teaching modules • Designing assessment tools • Examining students’ learning orientation (e.g., attitude, belief)

  11. 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。

  12. 培養學生數學素養的進路 Origins of Mathematics Within School Mathematics Beyond School Mathematics Common sense Experiencing the essence of mathematics learning

  13. 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 Taiwan Fou-Lai Lin Department of Mathematics, National Taiwan Normal University 2001 Conference on Common Sense in Mathematics Education: The Netherlands and Taiwan 19~23,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 methodology—developmental research and seek for suggestions from you, the developers of this methodology. My main focus will be on the learning of those new researchers.

  17. Ⅰ.The research program (08,2000~07,2005)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. 3.Children’s Informal Knowledge (I.K.) • A perspective of learning: Informal Bridging Formal Framework Knowledge

  21. Cognitive Architecture • eg. Defining a rectangle (long square in Chinese) INTENTIONAL AUTOMATIC The representationDENOTESthe representation the represented object in a : IS THE OUTCOME of a direct access to object discursive situations non-discursive situations non-aware memorized (reasoning operators) （visualization） implicit concept/theory facts images << Conceptualization >>

  22. 4.Focuses on I. K. and Concept Representation (1)Apprehension/ Recognization eg: • Apprehension of figure (Duval, 1995) • perceptual, sequential, manipulative and discursive apprehension • Recognizing a pattern(Bishop,2000) • Manipulative, proportional, recursive, functional approach.

  23. (2)Children’s Theory-in-action eg. Cognitive theory for practice(Vergnaud, 1998) :concept-in-action and theory-in-action Right angle Horizontal ^ Vertical

  24. (3) Intuitive Rules eg. The four intuitive rules(Stavy & Tirosh, 2000) More A-More B, Same A-Same B,… • Measurements(Chen, 2001) • Shapes(Chang,2001)

  25. (4) Visualization eg. Computer Models (ref: Tso,2001, this conference) (5)Representation eg. Function(Chang,2001)

  26. 5.Focuses on Cognitive Strategies • (Informal) Reasoning/Inferences eg. Exploring the definition and propositions of geometric shapes. (Lin, et. al, 2001) (2) Symbols eg. Linear equation (Wu, 2001)

  27. (3) Analogy eg. Defining a rectangle vs. defining a square. (Lin, et. al, 2001) (4) Proportionality eg. Recognizing a pattern. (Lin, et. al, 2001) (5) Modeling eg. Linear equation(Wu, 2001) Infinity(Wang, 2001)

  28. K 1 2 3 4 5 6 7 8 9 0 3 3 5 9 61 45 15 7 32 6.Focuses on Social-Cultural Cognition • Language eg. Percentage of students who responded that a square is a rectangle(long square in Chinese) Grader %

  29. (2)Indigenous Children eg. Criteria used to classify shapes • Ellipse 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

  30. (3)Superstition eg. Red-envelope and white-envelope vs. Even number and odd number • pronunciation of the digits 4;6;8. • Ten thousand dollars is called ’one dollar’(affective) • Is 1000 an even or odd?

  31. (4) Cognitive Styles eg. The Cram Industry • Learning by examples/Imitating • Practicing makes you skillful.

  32. 7.Infrastructure of the Research Program … … … …

  33. Ⅱ. A Researcher as a Learner: the Development of CD-MIT

  34. 1.Monday’s Seminar • Seminar on Mathematics Education Publications • Lasted for more than ten years • Kluwer’s Mathematics Education Library • Books from Frendental Institute (OW & OC; CD-B) • Studies in Mathematics Education Series, The Falmer.

  35. Others • Handbook of research on Mathematics Teaching and Learning. • Thought and Language; L. Vygotsky, MIT • Problems of representation in the teaching and learning of mathematics, LEA • Speaking Mathematically ,RKP • Mathematical Experience, and many others.

  36. Now, the seminar focuses on the book: Mathematics Education as a Research Domain: A search for identity.

  37. (2) Bi-Weekly workshops on the projects business • Clarifying • Presenting • Modifying • Re-designing • Making sense of the project

  38. 2.Two-days Workshop in each Semester • Reporting tentative results • Communicating the research

  39. 3.Regular Weekly Project Meeting ．Designing studies ．Reviewing Literature ．Analyzing data

  40. 4.Methods Used in the Program • Case based • Context based • Computer based • Clinical Interview • Pilot study with some classes in local region • Quantitative study, a national study with sample of about 1600 from each age population. • Teaching experiment

  41. Ⅲ.Conceptualizing Concept Development

  42. 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’)

  43. Participants: • Project directors(17) • Graduated students(4) • Teachers(2)

  44. 2.Data • 15 participants are able to describe their ideas during the workshop. • Various ideas about concept development

  45. 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) (Group Discussion)

  46. T1,2,3 (Ph.D Students) (Group Discussion) • Concerning the changes of the concept development： (1)Qualitative presentation:—a better control of the complexity of concepts (2)Strategy—more systematized when solving problems with concepts (3)Quantitative presentation—the facility of getting the right answers (4)The path of the concept development is not linear, is recursive.

  47. 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.

  48. It’s 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:

  49. 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

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