1. 1. TRANSITION 2. UNIFYING CONTENTS 3. LANGUAGE 4. NEW TECHNOLOGIES 5. CREATIVITY 6. TESTING 7. SOME CONTRADICTIONS 8. WINDS OF WAR 9. THE ISSUE OF GEOMETRY 10. AIMS AND GOALS 11. SOME DATA WG 2: MATHEMATICS EDUCATION IN JUNIOR SECONDARY SCHOOL Ferdinando Arzarello (Italy), CO Alwyn Olivier (South Africa), CO Rick Billstein [USA], AO Keiichi Shigematsu [Japan], AO Suwattana Utairat [Thailand], AO Nanae Matsuo, LAO IPC Liason Member: Julianna Szendrey [Hungary]

2. MORE THAN 100 PARTECIPANTS 3 HALF AN HOUR LECTURES 5 TWENTY MINUTES PRESENTATIONS 1 WORKSHOP 160 MINUTES DISCUSSIONS (3 SUBGROUPS AND A PLENARY ONE) WG 2 A brief statistics 1. Interesting issues 2. Interesting research questions 3. Which developments since last ICME 4. Where might the WGA go on the next ICME 5. Which obstacles to the implementation of new ideas THE ITEMS SUGGESTED BY IPC

3. MORE THAN 100 PARTECIPANTS 3 HALF-AN-HOUR LECTURES 5 TWENTY-MINUTES PRESENTATIONS 1 WORKSHOP (50 MINUTES) 160 MINUTES DISCUSSIONS (3 SUBGROUPS AND A PLENARY ONE) WG 2

4. THE MAIN PROBLEM: TRANSITION WG 2

5. ICME 9 School education from grade VI to grade X is characterised by TRANSITIONS:  from compulsory education to jobs and/or vocational school education;  from school-for-all to different schools (or curricula).  from child to adult behaviours of learners within the 11-16 age range.

6. ICME 9 Mathematics education plays a special function in these transitions:  it must ensure the acquisition of crucial tools for life and many professions;  it must prepare students for further studies in scientific and technological areas;  it is supposed to pass over relevant components of modern scientific culture.

7. ICME 9 The problem of transition has been investigated and discussed from different points of view, e.g.: MATHEMATICAL COGNITIVE CULTURAL POLITICAL

8. ICME 9 From a MATHEMATICAL point of view, it has been underlined the necessity of introducing important and challenging issues, which appear as UNIFYING and crucial in the present historical moment (which is itself also featured by deep transition -social and political- aspects).

9. ICME 9 From a CULTURAL point of view: the learning of mathematics evolves differently in different countries, because of cultural reasons.

10. ICME 9 From a POLITICAL point of view, the transition has been considered underlyning: * The evolution of the school organization. The specific tendencies and the general features of the teaching of mathematics in JSS in different countries have been considered. * The suggestions mathedu research can give to politicians in order to improve mathematics teaching in the school.

11. 1. TRANSITION7. SOME CONTRADICTIONS 2. UNIFYING CONTENTS8. WINDS OF WAR 3. LANGUAGE9. THE ISSUE OF GEOMETRY 4. NEW TECHNOLOGIES10. AIMS AND GOALS 5. CREATIVITY11. SOME DATA 6. TESTING WG 2 WITHIN THIS FRAMEWORK THE FOLLOWING ISSUES HAVE BEEN DISCUSSED:

12. WG 2 ISSUES RESEARCH NEWNEXTOBST. TRANSITION UNIF. CONC. LANGUAGE TECHNOL. CREATIVITY TESTING CONTRADICT MATH WAR GEOMETRY AIMS,GOALS NUM. DATA          

13. WG 2 SIMPLE ARITHMETIC INDUCTION INTUITIVE PROCESSES COMPLEX ALGEBRA DEDUCTION PROOF PRODUCTS 1. TRANSITION

14. WG 2 2.UNIFYING CONCEPTS MATH COGNIT CONTENTS PROCESSES DIDACT.

15. WG 2 2. UNIFYING CONCEPTS MATH COGNIT CONTENTS PROCESSES DIDACT. A) PROCESSESB) CONTENTS C) COGNIT.

16. WG 2 A) PROCESSES vs/PRODUCTS CONTENTS asACTIVITIES &CREATIVITY

17. WG 2 B) CONTENTS: Which? How much? Functions (linearity) Algebraic thought

18. WG 2 C) FLEXIBLE PATTERNS THROUGH OPEN PROBLEMS SISTEMATICITY ?

19. WG 2 3. LANGUAGE INNER PROCESSES LOGICAL SKILLS COMMUNIC. READING & WRITING

20. COMMUNICATION It is related with: processes culture Pupils express themselves Helps students to set down their working into a framework Differences between pupils/teacher/book WG 2

21. 4. TECHNOLOGY + DIFFICULT PROBLEMS LOGICAL SKILLS - NUMERACY ? FINAL EXAMS (paper&pencil) DGS: 3D INTUITION  PROOF CALCUL., PC & WEB TEACHERS’ TRAINING FAMILIARITY OF CHILDREN WITH NT: IT VARIES A POLITICAL PROBLEM

22. WG 2 HOW TO DEVELOP IT ? DEFINITIONS: ideological features cultural issues 5. CREATIVITY

23. WG 2 6. TESTING OPEN ENDED VS/ STRUCTURED PRODUCTS VS/ PROCESSES

24. WG 2 7. SOME CONTRADICTORY ISSUES CHALLENGING THE INNOVATION CHALLENGING THE TRADITION ? BUT !

25. WG 2 CHALLENGING THE INNOVATION  OPEN ENDED MATHS problem solving with open, non routine problems (possibly, no systematic learning)  INTEGRATED LEARNING (basic skills in a more or less traditional way)

26. WG 2 CHALLENGING THE TRADITION Challenging the stability of mathematical thinking development in students Example: hierarchies vs/ networks

27. WG 2 BUT IN CHILE, JAPAN,… SOME OF THE CHALLENGED INNOVATIVE ISSUES ARE AT THE BASIS OF THE NEW REFORMS OF THE SCHOOL

28. WG 2 8. WHAT DO THEY MEAN THESE WAR WINDS THAT ARE NOW BLOWING?           

29. WG 2 There are elements which are difficult to construct in a constructivist approach to theoretical knowledge and difficult to mediate through a traditional approach.

30. WG 2 9. THE ISSUE OF GEOMETRY USEFULNESS FOR: 3D  2D conj.  proof A BASIC PATH IN SOME COUNTRIES (but numerical skills and algebraic methods are overemphasized in others) NATURAL LANGUAGE

31. WG 2 10. AIMS AND GOALS: COGNITIVE (understanding, math literacy) METACOGNITIVE (thinking in a developmental way; ways of looking positively at maths)

32. WG 2 THE LAST THREE QUESTIONS of IPC WHAT NEW FROM LAST ICME? WHICH NEW RESEARCHES FOR NEXT ICME? WHICH OBSTACLES TO THE IMPLEMENTATION OF NEW IDEAS?

33. WG 2 WHAT NEW FROM LAST ICME? THE MATHEMATICAL WAR  AND THE NECESSITY OF A 3RD ROUTE IN BETWEEN CONSTRUCTIVISM AND TRADITIONAL METHODS

34. WG 2 NEW RESEARCH ISSUES FOR ICME 10 1. WHICH EXPERIMENTAL EVIDENCE THAT COMPUTERS, WEB,... DO INCREASE MATH LEARNING ? WHAT AND HOW TO ASSESS? 2. TWO DIFFERENT APPROACHES TO TRANSITION: a) CONTENTS-ORIENTED b) SYSTEMIC-ORIENTED

35. WG 2 3. AIMS AND GOALS ARE CULTURALLY DEPENDENT; USEFUL A COMPARISON: BETWEEN STATES WITHIN A STATE (DIFF. SUBJECTS) 4. FOCUSING TRANSITION FROM ELEMENTARY AND TO HIGHER SEC. SCHOOL. 5. CONSEQUENCES OF TESTING IN DIFFERENT COUNTRIES.

36. WG 2 OBSTACLES THE LACK OF TECHNOLOGY FEABLESS OF TEACHERS TRAINING IN NT

37. WG 2 11. SOME DATA: # STUDENTS: 20 - 60 FOR EACH CLASS # HOURS PER WEEK: 2.5 - 5 DIFFERENT AGES OF STUDENTS DIFFERENT AIMS