1. Teacher Training in Japanese Mathematics Education Toshikazu Ikeda Yokohama National University Yoshiaki Kuwahara Shinmei Junior High School

2. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Outline States and issues in Japanese teacher training Pre-service education : Example in Yokohama National University In-service education Lesson Study

3. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara First Report by the Education Personnel Training Council (1997) Improvement of the teacher training curriculum ① Wide understanding of liberal arts ② Problem Solving Ability, Humanity ③ Knowledge and skills of subject matters The number of credits for subjects concerning teaching methods of mathematics → From about 2 credits to 8 credits

4. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Second Report by the Education Personnel Training Council (1998) Promotion of reeducation of currently employed teachers through master course System of night master course for teachers is gradually introduced in teacher training university

5. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Third Report by the Education Personnel Training Council (1999) Facilitation of cooperation in the cultivation, appointment and training of teachers Deeper cooperation is expected University Board of education School

6. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Issue in teacher training university In 2001 Ministry of Education, Culture, Sports, Science and Technology is planning to integrate teacher training university, keeping the quota of 10,000 students. Pre-Service Education AllNothing

7. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Example in Yokohama National University In 1997: Faculty of Education → Faculty of Education and Human Sciences ◇ Pre-service education: 230 students Focus on Elementary School Education ◇ Establishment of New Courses: 230 students Quota of students 15000 → 10000 In 2000: Introduction of a New Curriculum based on first report by the Education Personnel Training Council

8. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Aims in mathematics education in Yokohama National University [Mathematical points of view] ① To foster students’ ability to think mathematically ② To enable students to understand mathematical concepts and ideas existing behind elementary and secondary mathematics. ③ To foster students’ ability to represent mathematically ④ To foster students’ ability to utilize technology

9. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Aims in mathematics education in Yokohama National University [Educational points of view] ⑤ To enable students to understand the objectives and contents of mathematics ⑥ To enable students to understand Teaching methods including assessment of mathematics ⑦ To foster students’ ability to construct and analyze a classroom teaching ⑧ To foster students’ ability to utilize the textbook of mathematics

10. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Introduction to Math. Observation of Classroom teaching Calculus Ⅰ - Ⅳ Algebra Ⅰ - Ⅳ Geometry Ⅰ - Ⅳ Probability & Statistic Ⅰ - Ⅳ Teaching Practice [elementary level] (4 weeks) Teaching Practice [secondary level] (2 weeks) Teaching methods of elementary math. Problem solving and mathematical thinking Analysis of Classroom teaching Contents taught in Secondary math. Development of Teaching material Assessment Lesson plan & observation 1 2 3 4 Report to graduate Curriculum of mathematics education in YNU

11. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara In Case of Yokohama National University (For Elementary School Teacher) Introduction to Mathematics for elementary school teachers Mathematical concepts and ideas existing behind elementary mathematics １． Number and operation ２． Quantity and measurement ３． Figures ４． Relation between number and quantity ５． Way of mathematical thinking Core １． Mathematics for teachers ２． Mathematics for a real world ３． Mathematics for a joy Option Presentation technology

12. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Option: Mathematics for a real world Presentation, Assessing each other Mathematical points of view: 5points Presentation: 5points Selection of problem in each group 12 groups composed of 5 members Problem Solving Preparation of Presentation

13. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara GOAL KICKS IN RUGBY In rugby, a chance to score a goal is given after a try. The probability of a successful goal is influenced by the position where the goal kick is made. How should you decide the position with the largest probability when attempting to make a goal kick. ＴＲ Ｙ Goal Post It is possible for player to kick in any place on the blue line.

14. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara “How many years does it take to make money double?” ５％ → ７２ ÷ ５＝１４．４ １ ５ years ３％ → ７２ ÷ ３＝２４ ２ ４ years １％ → ７２ ÷ １＝７２ ７ ２ years Law of 72

15. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Eyesight I heard that Eyesight of Inuit is more than 5.0. If it is true, how can Inuit see far away compared with us.

16. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Mirror Problem Problem What size of mirror do we need at least in order to see the whole body?

17. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Reflector of bicycle Reflector of bicycle is shining even though light flood into it from any angles. What is the structure of reflector?

18. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara In-service teacher training in Japan

19. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Classification of Teacher Training Public Training (National,Prefecture,City/Town/Ward, School) Private training (Individual research,society,Other Training & research circle)

20. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Public Training for in-service Teachers New teachers Training Experienced Teachers Training (After 5 years & After 10 years ） Dispatched Training (Master course in teacher training university, Company, Abroad ）

21. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara New teachers Training ・ About 9 ０ times training during first year 60 times in School, 30 times out of School ・ Leading Teacher in charge guide New Teacher one by one. ・ Extensive training Lecture,Lesson study,Management

22. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Experienced Teacher Training After 5 years ５ ~ ６ times training in 6th years Group Training in each theme (Class Management, Students Guidance, Problems to be solved (Bullying etc) After 15 years 3~5 times training in 15th years Optional Course Training each interest (Subject,Class Management, Students Guidance, Problem to be solved ）

23. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara An example in mathematics education In case of Junior High School Teachers Contents: ・ Explanation of the National Curriculum ・ Developing New teaching materials for Team Teaching or Optional Course ・ Collaboration by Experienced Teachers

24. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Dispatched Training To a Company To Abroad Language Training Japanese School Inspection of various countries To a Master Course in teacher training university (Study & Research)

25. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Collaboration on the Daily work Daily discussion in the Teachers ’ Room Regarding to mathematics teaching Progress of classroom teaching The way of Assessment Teaching Materials Exchange views

26. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Lesson Study as teacher training Pre-service education: One of the tools to assess a classroom teaching Examinations to become a teacher: Demonstration of classroom teaching ① Discussion about a lesson plan ② Observation of an actual classroom teaching ③ Analytical discussion about the classroom teaching

27. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Lesson plan – it’s history Teacher’s activity Students’ activity Remark on teaching Before world war Ⅱ After world war Ⅱ

28. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Five styles of a Lesson Study in Japan (in-service education) Scale of L.S. By Whom Style 1 In a school Principal & Teachers Style 2 In a City/Prefecture Teachers themselves Style 3 In a City/Prefecture Board of Education Style 4 From all of the prefectures School attached to University Style 5 In a Prefecture, From all of the prefectures Academic Society & Company

29. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Five styles of a Lesson Study in Japan (in-service education) Main Aim of L.S. Methods to disseminate Style1 to attain the school objectives Announcement Style2 to promote professional development Convey from older teachers to younger teachers Style3 to develop and spread new contents and approaches Call for attendance at the beginning of semester Style4 to develop and spread new contents and approaches Announcement ， Homepage Style5to cultivate activity of lesson study Announcement, Journal, Homepage

30. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Relation between children and teacher Relation between teacher and material Relation between children and material Strong Relation Between two components Lesson Study Triangle children material teacher

31. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Effectiveness of Lesson Study Lesson study might be effective for teacher in the following points in particular. ① learn to see material ・ Can teacher identify mathematical essential points of materials? ・ Does teacher deprive of students ’ opportunity to think mathematically?

32. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Effectiveness of Lesson Study ② learn to see students ・ Can teacher understand what students understand? ・ Can students understand teacher ’ s asking questions? ・ Does teacher ignore of students ’ ideas by his/her selfish reason? ・ Can teacher accept and evaluate students ’ ideas appropriately? ・ Can students discuss cooperatively?

33. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Effectiveness of Lesson Study ① learn to see material + ② learn to see students ・ Can teacher develop the material by considering both mathematical essential points and students’ interest? ・ Can students find out their own problems from the original problem that teacher presents at first? ・ Can teacher assist students’ cooperative discussion and summarize the mathematical essential ideas derived from their discussion?

34. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara What do you consider important when introducing a mathematics lesson? (3) To make students clarify an ambiguous situation. Answers of students Why did we derive a variety of ideas ? Students’ own problemCheck 3 in the following figure !

35. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Learn to see material Identifying mathematical essential points To identify essential and a wide range of mathematical ideas Ex: Division with a remainder “13÷5 = “ [ Aim of teacher ] The quotient is 2 and the remainder is 3 If I had two more, the quotient would become 3 The quotient is 3 with a lack of 2 No, your idea is not good.

36. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Learn to see material Identifying mathematical essential points Wide and deep mathematical insights are required for a teacher. Ex: Division with a remainder “13÷5 = “ Mathematical essential points [ Aim of teacher ] The quotient varies based on the situation Teacher can pick up students ’ mutter as well as focus on mathematically essential points.

37. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Let’s measure the weight of three animals to compare the volume Which animal do you think is the biggest among a lion, a frog and a turtle made by the clay ? Um… Are there other ideas ? It is not appropriate for teacher to ignore student ’ s idea in this situation. Is there any students’ ideas ignored by teacher’s selfish judgment? Focus of teacher: Measuring Volume

38. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara 15, July Are mathematical essential ideas elicited from students’ collaborative discussion summarized at the end of lesson? Students developed the formula to calculate the area of trapezoid. At the end of lesson, teacher put the panel on the blackboard. Summary S = (a+b)× ｈ ÷ ２ a b h If you have the panel, please show us at the beginning.

39. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Future Issues to be discussed in Japan It is necessary for teachers to discuss the issues that are useful in a future teaching. Namely, it is expected for teachers to discuss the following questions for instance ． Is there any alternative teaching methods? If you have a chance to teach the same topic next time, how do you want to treat them? What is the points that is useful in a future teaching?

40. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara Future Issues to be discussed in Japan We often observed the stereotyped classroom teaching. (ex. Understand a problem - solving individually - Presentation by students - Negotiation) The most important thing for teacher is to accept students ’ ideas and evaluate them flexibility. If a teacher couldn ’ t do so, students might not come to life.

41. PCMI 2002: Toshikazu Ikeda, Yoshiaki Kuwahara

42. The number of new teachers & the number of teachers Total 12,000 Total 31,000 Elementary:407,000 (Male):(Female) =2:3 Junior High: 255,000 (Male):(Female) =3:2 Senior High: 266,000 (Male):(Female) =5:2 Number of Teachers in 2002