1. 1 Developing Deep Understanding of Mathematics Teaching LI Shiqi East China Normal University sqli@math.ecnu.edu.cn

2. 2 Outline §Background: General situation in China §Challenges: How to develop deep understanding of assessment in teaching l What to be focused in assessment? l How to assess? l Who do the assessment? §Summary

3. 3 General situations of teaching research activity (TRA) in China There is a long history of teaching research activity in China There is a long history of teaching research activity in China Originally, it was a school-based teaching exchanges, but now it is extended to national wide academic activity Originally, it was a school-based teaching exchanges, but now it is extended to national wide academic activity There are some new trends developed There are some new trends developed

4. 4 TRA structure: levels of TRA §Parallel teaching group at every grade §Teaching research group at school §School district §Province / city level §Area level (Northeast, East China, etc.) §National level, organized by national academic societies or associations

5. 5 Focus of TRA: §Lesson planning §Teaching suggestions §Lesson observation §Discussion and reflection § … …

6. 6 Aim: Improve practical teaching, §Teaching objects, §Global Structure §Steps and procedures, §Teaching behaviors, §Students response & achievement then encourage education research

7. 7 Forms of teaching research: §Open lesson ( ) §Model lesson ( ) §Research lesson §Teaching competition §Lesson explanation ( )

8. 8 Challenges: How to develop deep understanding of assessment in teaching §What? pedagogy focused / mathematics focused pedagogy focused / mathematics focused §How? qualitative / quantitative qualitative / quantitative §Who? experts / teachers experts / teachers

9. 9 What: pedagogy focused/math focused Belief: Math ideas and principles are the heart of math lesson. Between math and pedagogy, correct math is always put in the first place. Teacher must pay good attention to the math understanding and suitable treatment of teaching material. Some cases of teaching: l Teaching Sine Law with exploration l Situated teaching l Midpoint connectors: a teaching aid l Some evaluation forms

10. 10 Case 1: Teaching with exploration Case 1: Teaching with exploration Process of teaching of Sine Law Students were grouped and draw own triangles, measured its angles and sides; then computed some data such as: c/sin C a/sin A, b/cos A, etc. Students were grouped and draw own triangles, measured its angles and sides; then computed some data such as: c/sin C a/sin A, b/cos A, etc. Some students report their results and fill them in a form Some students report their results and fill them in a form

11. 11 Group abc A B C c /sinC b /cosA a /sinA a /conB b/ sinB A 4.13.33.7570 0 50 0 60 0 4.3309.6494.3636.3784.308 B 5.33.13.6107.5 0 33 0 39.5 0 5.660- 10.3015.5576.3205.692 C 33360 0 2.5986 6 Some data from students group Some data from students group

12. 12 Doubt: Is there any vital problem in the process of teaching design? Following teaching steps: Teacher let students make conjecture, and he wrote the correct conjecture on blackboard Teacher let students make conjecture, and he wrote the correct conjecture on blackboard Next step: Teaching on to apply the law Next step: Teaching on to apply the law

13. 13 A B C Doubt: How to set situation for teaching? Case 2: Situated teaching Case 2: Situated teaching The minimum distance for fire fighting The minimum distance for fire fighting

14. 14 Case 3: Introduce the concept of midpoint connector of trapezoid from the one of triangles

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18. 18 Making connection between concepts !

19. 19 Case 4: Some improvements of indicators in evaluation form for lesson observation §Form 1 §Form 2 §Form 3

20. 20 How: qualitative/quantitative Let qualitative and quantitative messages send suitable implications to teachers §A paper: Insight into mathematics teaching

21. 21 Case: A quantitative ways of analysis: Questioning analysis A Administrative Questioning A Administrative Questioning B Mechanist B Mechanist C Remembering C Remembering D Explanative D Explanative E Reasoning E Reasoning F Criticizing F Criticizing

22. 22 Questioning Analysis A. Administrative Questioning A. Administrative Questioning Who has any new ideas about it? Who has any new ideas about it? B. Mechanist B. Mechanist How many auxiliary line are there? How many auxiliary line are there? C. Remembering C. Remembering How did we prove it last time? How did we prove it last time?

23. 23 Questioning Analysis D. Explanative D. Explanative What is base side and what is the third side? What is base side and what is the third side? E. Reasoning E. Reasoning Why do you draw such a auxiliary line? Why do you draw such a auxiliary line? F. Criticizing F. Criticizing Why this is a wrong way? If so, what is your new suggestion? Why this is a wrong way? If so, what is your new suggestion?

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27. 27 Questioning comparison

28. 28 Complicated questioning Complicated questioning

29. 29 Simple questioning Simple questioning

30. 30 Important behavioral differences between two teachers Mr. A Mr. B Definition introduction At the beginning After proof Proving Just proving directly From Conjecture to proving Situated problem As application of theorem As the introduction to theorem Knowing theorem Reciting Read text Rephrasing theorem Word by word same as on text Right but flexible How many … How many … Tell to students Hint The difference to median Tell to students Hint Writing on chalkboard FormallyOutline Didactics principle Thoroughly, deeply and clearly explain Less explain and more practice

31. 31 Who: expert centered/teachers centered §A characteristic: teaching researchers play an important role §Change the pattern of Teacher teaching and experts comment: Lesson explanation: self description and reflection (Huang) §Online learning and assessing by teachers §Yang: interesting research result: 3 rounds action learning not so good as expected

32. 32 A new trend: lesson explanation A new trend: lesson explanation Teachers explain and reflect his/her own design of a lesson, its underlining ideas and related theories An example: Dr. HUANG Xinfengs work: The sum of the first n terms of an arithmetic series

33. 33 For a general view, please read: Peng, Aihui (2007): Knowledge growth of mathematics teachers during professional activity based on the task of lesson explaining, Journal of Mathematics Teacher Education, 10: 289 – 299

34. 34 Another kind of teachers reflecting activity: Online learning and assessing in Shanghai Videotaped lessons are put online every three months or so. Teachers are required to observe and write own comments and questions online as a course work. Videotaped lessons are put online every three months or so. Teachers are required to observe and write own comments and questions online as a course work. Teaching researchers will read such course work and send response to them. Teaching researchers will read such course work and send response to them. Every teachers who finish the work will earn their training credits. Every teachers who finish the work will earn their training credits.

35. 35 A Case: Experts special research will give teachers more insights into practical teaching YANG, Yudong (2005): Classroom Teaching Driven by Primitive Mathematics Ideas An Action Research for Improving Mathematics Teaching, Journal of Mathematics Education (In Chinese), 14(2), 59-63.

36. 36 Interesting finding: three rounds of teaching improvement not so perfect as expected §First round: teachers planed lesson and teach it himself there are some weaknesses §Second round: teachers improve their teaching with more comments and suggestions from experts etc. even less successful than the former one §Third round: teachers reflected their experience independently, adjust their lesson plan and teach again it seemed better and more successful

37. 37 Summary: Complementary & interdependent ways make lesson study assessing effective §Pay attention to both mathematics & pedagogy: keep right track of math teaching carefully §Apply both qualitative and quantitative evaluate ways: reveal and insight into the keys of teaching §Both experts and teachers do teaching assessment: improve practical teaching effectively

38. 38 Thank you for your attention !