Fou-Lai Lin and Chuang-Yih Chen Department of Mathematics National Taiwan Normal University

1. ConceptualizingMathematics Concept Development :Mathematicians and Maths Educators as Co-learners Fou-Lai Lin and Chuang-Yih Chen Department of Mathematics National Taiwan Normal University Plenary Speech on “International Conference on Science & Mathematics Learning” December 16-18, 2003 , Taiwan

2. The Context An Integrated programConcept Development:Mathematics in Taiwan (CD-MIT) (08,’00~07,’03)

3. The Topics Studied

4. The Topics Studied *8/17 P.I are Novice researchers in Math. Ed

5. Schedule Stage 0 Stage 1 Stage 2 Aug,’00 Nov.’01 Jan~May ’02 Nov.’02 Aug.’03 Prog. Organizing interview pilot study national final with P.I survey report Data1 Data2

6. Stages of Understanding Children’s Mathematics Concept Development

7. Conceptualizing • Views about C.D at Stage 1 (Data 1) vs. Performances at Stage 2 (Data 2) • Qualitative analysis • Driving Force • Learning Process • Learning Product

8. Hypothetical Learning Routes for Maths. Eds on C.D. • H.L.R • R1:stage 1, 2, 3, 1, 2, 3,...4, • R2:stage 1, 2, 3, 4 • R3:stage 1, 2, 1, 2, 3,...4, • Reflection • Learning shows the insufficient 學然後知不足 • Knowing the insufficient 知所不足 (Confucisous)

9. The End~ Thank You

10. Views about C.D. at Stage 1 R • Remainder ( degeneration) eg.T11 • Linear eg.T4 • Horizontal ; eg.T7 ,T12 • Navigating • Expanding • Inductive • Vertical ; eg.T8 , T10, T13, T14 ,T15 • Hierarchy • Internerization • Sfard’s theory • Accommodation • Social Interactive ; eg. T1,2,3 (Theoritically),T6 • Language & representation L H V S

11. Views (Stage 1) v.s. Performance(Stage2) Views Performances

12. Educating Mathematics Educators Q1:Four years ago, what were your research areas? Maths., statistics, computer, edu., maths. Edu. Q2:Would you continue to research mathematics education? N=17 P.I (Dec. ‘03)

13. Data 1 Interview: • Would everyone explain your ideas about concept development? (05,11,’01) • Subjects: -Project directors (17) -Graduated students (4) -Teachers (2)

14. ( Dual ; Social interaction ; ) • 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)

15. Concerning the changes of the concept development： T1,2,3 (Ph.D Students) (Group Discussion) (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.

16. T4 ( Process concept ; Readiness ; ) • 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.

17. ( ; CD , Inductive ; Computer model ) 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:

18. T6: ( Def., S.I.R ; Functional ; Midwife, Sculptor) (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

19. T7: (Exp,Non-exp. ; Navigating ; ) (1) It’s navigating 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)

20. T8: ( Misconception, Language ; Accommodation ; ) (1)The intuitive understanding is from semantic interpretation of the words. (2)Then accommodation by the conflicts and counterexamples towards the final concepts. (3)For example, independent events is intuitively viewed as disjoint events…

21. T9: ( Acceptance ; ; ) (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.

22. T10: ( Set of subconcepts; Text-book based ; ) • 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.

23. T11: ( Remainder of cramming ; Remaining process ; Barred Island ) • 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 one’s mind, it is not easy to forget. • This metaphor could be used to build up the hierarchy among concepts.

24. T12(1): ( Manifold ; Change, Inductive ; City development) • Use the concept ‘Development’ as an example of concept. • Interpretation the concept ‘Development’ for example • Chicken is growing • Duck is growing • Dog is growing Chicken and duck both are oviparous. Dog is viviparous. They are changing from samll, hairless to big and with hair.

25. T12(2): ( Manifold ; Change, Inductive ; City development) • 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.

26. T13:(1) ( Manifold, S.I.R ; Smoothing ; Envelop model) • 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 don’t have its definition. • It can be explained by the envelop model( envelop of curve, surface)

27. T13:(2) ( Manifold, S.I.R ; Smoothing ; Envelop model) • A concept is enveloped by • Features of the concept • Situations • terminology • Symbolic representation • Examples • Under 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

28. T14: ( Informal vs. Formal ; Informal to formal hierarchy ; understanding level ) (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.

29. T15:(1) ( Specific, Accurate, Economic ; Accommodation cognitive strategy ; ) (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’ ‘look the same’ ‘proximate’ ‘akin’, etc.

30. 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.

31. T4’ Performance on C.D • Intuition • Formal learning • Hierarchy Comment: L

32. T6’ Performance on C.D • Hierarchy (situations, features) • Architecture of development (attributes, features) Comment :V

33. T7’ Performance on C.D • 2-dim. Specification table • Strategy • H.L.U.R Comment: H

34. T8’ Performance on C.D • Hierarchy (situation) • Formal learning Comment: L

35. T10’ Performance on C.D • Ambiguity of Tolerance • Condensation ( representations, examples) Comment: H

36. T11’ Performance on C.D • Attributes • Proto-type • Over-generalization • Strategy of proportional reasoning • Back to semantic Comment: H , R

37. T12’ Performance on C.D • Strategy • Situation • Diagnostic assessment • Representation Comment: V

38. T13’ Performance on C.D • Attribute and Strategy • Difficulty Comment: H

39. T14’ Performance on C.D • Hierarchy • Strategy • Misconception Comment: V

40. T15’ Performance on C.D • H.L.T • Effect • Hierarchy Comment :V

41. Activity and Driving Force By Stage

42. Process and Product by Stage

43. Driving Force at Stage 0 Why the learning community organized? Motives of joining the program： Atmosphere of the Campus in 2000： • Responsibility of university faculty. Teaching, Research, Service • Mission of the Institutions. • Self-Regulation

44. Driving Force at Stage 1 • Two Dimensional Specification. • Pilot Study

45. Working on a two Dimensional Specification. • Dimension X: Features and Attributes of the Topic. • Dimension Y: Context of Concept Development and Cognitive Strategy.

46. Two Dimensional Specification

47. Pilot Study • Interview • Developing coding • Conjecturing

48. Processes at Stage 1 • Chaos exploring • Clarified with 2-dim specification. • Synthesizing as a booklet

49. Products at Stage 1 Ⅰ.Rich Information on the 2-dim specification table. Ⅱ.Test Booklet Ⅲ.Hypothetical Local Understanding Route.

50. Driving Force at Stage 2 • National Survey 2. Discourse within the community • Leadership • Narrative • Reflexion