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The Teaching Of Mathematics At Senior High School In France
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The Teaching Of Mathematics At Senior High School In France

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2009年4月21日下午,厦门大学嘉庚三215会议室,数学教育沙龙。

2009年4月21日下午,厦门大学嘉庚三215会议室,数学教育沙龙。

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  • I would like to thank the organizers of this conference, and especially Professor Zhang Yingbo for their kind invitation at this session on mathematics education. It is for me a great pleasure to exchange with Chinese experts in that area, and to try to understand better through these exchanges your educational culture which is renowned all over the world. I have been asked to spreak about the teaching of mathematics at senior high school in France. I will do it in the following way.

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  • 1. Michèle Artigue, Université Paris Diderot - Paris 7 President of ICMI THE TEACHING OF MATHEMATICS AT SENIOR HIGH SCHOOL IN FRANCE
  • 2. Summary
    • The structure of secondary education in France
    • Senior high school Education in France today: the challenges to face
    • The 2000 Senior High School Reform: ambitions and curricular choices
    • The 2000 High School Reform: the implemented curriculum
    • Ten years after: Towards a new curricular organization?
  • 3. The structure of French Secondary Education
  • 4. French secondary education: the global structure Middle school (4 years - from age 11 to age 15)
    • General or Technological high schools (3 years):
    • Seconde
    • Première
    • Terminale
    • Vocational high school (2 to 4 years):
    • BEP (2 years)
    • Vocational Baccalauréat (2 years)
  • 5. French secondary education: the general high school
    • Since 1991, three main orientations exist in general high schools:
      • L (Literature, languages, philosophy and arts)
      • ES (Economy and social sciences)
      • S (Sciences)
    • Differentiation between orientations begins in Première and reinforces in “Terminale” with the choice of a “Specialty” (Maths, Physical sciences or Biology for the Scientific orientation)
    • There is a national syllabus for each grade and orientation; textbooks are freely produced by editors and selected by schools
  • 6. The French secondary educational system : maths time-tables Level Orientation Time-table Mini-Maxi Seconde   C:3h – TD*:1h AI*:1h 4h - 5h Première L C:1h - TD:1h Optional:3h 2h - 5h Première ES C:2,5h TD:0,5h Optional:2h 3h - 5h Première S C:4h - TD:1h 5h Terminale L Optional:3h 0h - 3h Terminale ES C:4h Optional:2h 4h - 6h Terminale S C:4,5h - TD:1h Optional:2h 5,5h - 7,5h
  • 7. The main challenges to face
  • 8. Senior high school education: The challenges to face
    • The massification of senior high school education
    • The necessary adaptation to the mathematics and sciences evolution and to that of social and professional needs
    • The necessity of copying with the technological evolution
    • The disaffection for scientific careers
    • All of this in a context of increased competition with other disciplines, and of reduction of math hours
  • 9. The 2000 High School Reform
  • 10. The 2000 High School Reform
    • A reform piloted by the CNP and designed by a mixed group of experts including mathematicians, secondary teachers, mathematics educators and inspectors. Its chair was Claudine Robert, a statistician.
    • Not a revolution but some important curricular changes, and among these:
      • an increased place given to statistics and probability
      • an increased differentiation in the curriculum according to the different orientations
      • an increased importance attached to the links with other scientific disciplines
      • an increased role given to technology beyond calculators (especially dynamic geometry software and spreadsheets)
  • 11. The main reasons underlying these changes
    • Taking into account the evolution of mathematics, of the relationships between mathematics and other scientific disciplines, and the evolution of social mathematical needs
    • Trying to overcome the vision of the scientific orientation as the “excellence” orientation, and its associated negative effects, paying better attention to the diversity of mathematics professional needs
    • Developing students’ autonomy, ability to perceive the role that mathematics play in the surrounding world, and to work collaboratively with others
    • The progressive deterioration of coherence due to the successive adaptations of the syllabus, and the exaggerate influence on teaching practices of emblematic tasks of the baccalauréat
  • 12. Statistics and Probability
    • Statistics is one of the three main domains in the syllabus for the « Seconde » level (1/8 of the time)
    • Statistics is not limited to descriptive statistics but includes some initiation to inferential statistics through the study of sampling fluctuations
  • 13. Going beyond descriptive statistics…
    • « The statistical spirit is born when one becomes aware of the existence of sampling fluctuations […] The teaching choice which is made is going from observation towards conceptualisation, and not of introducing first the probabilistic language to notice then that all occurs as the theory envisages it  »
  • 14. Topics for study linked to statistics
    • Simulation of a survey, with samples of variable size, confidence intervals associated with a survey and associated formulas
    • Simulation of pile or face game
    • Simulation of the throw of two identical dices and distribution of the sum of the faces
    • Simulation of random walks on solids or polygonal lines, time fluctuation and estimates of average time used to go fom a given vertex to another one
    • Simulation of births and distribution of the number of children, boys, girls per family under specific conditions (4 at most and stopping at the first boy for instance)
  • 15. Probability
    • Probability teaching begins in « Première » and goes on in « Terminale »
    • It includes approach to discrete and continuous laws
    • Its ambition: « to see in simple cases what a probabilistic model is and to approach probabilistic computations »
  • 16. The links with other scientific disciplines
    • The multidisciplinary projects (TPE) introduced in « première »
    • The introduction of some multidisciplinary work around radio-activity leading to a new mode of introduction of the exponential function in « terminale S »
  • 17. What are the TPE?
    • Students, working in small groups, have to develop a collective project over one semester, leading to a real production
    • TPE must bring into play at least two different disciplines, among these a major in the students’ orientation
    • 2 hours per week are allocated to the TPE preparation and these are jointly supervised by teachers of different disciplines
    • The precise theme of the project is chosen by the students themselves but has to be related to a list of national themes, and approved by the teachers supervising it
    • Students defend their project and production orally (jury of 3 members)
  • 18. The aims of TPE as presented in official documents
    • P roviding students with the opportunity of developing a multidisciplinary approach to questions which are not just school questions
    • H elping them to mobilise their academic knowledge in such a context,
    • W idening their intellectual curiosity
    • D eveloping their autonomy
    • H elping them to acquire methods and the competencies required for working in groups
    • D eveloping the abilities necessary for an effective search, selection and critical analysis of documentary resources, including Internet resources
    • Establishing more open relations between teachers and students
  • 19. Exponential: a new introduction
    • Introducing the exponential functions as solutions of the differential equation f’=kf
    • Linking this equation to a probabilistic modelling of radioactive desintegration:
      • The probability F(s) for a nucleus of desintegrating between 0 and s, is the same as its probability of desintegrating between t and t+s, for any t and s.
      • Thus G=1-F verifies G(t+s)=G(t).G(s) and is an exponential function
      • By using a binomial distribution and passing to the probabilistic expectations, one obtains the expectation of the number of nucleus at time t, starting from N(0) nucleus at time 0: N(t)=N(0)G(t)=N(0)e -  t
  • 20. A fundamental topic revisited: elementary analysis (calculus)
    • Since the seventies, a secondary calculus in scientific orientation rather ambitious trying to built analysis on the idea of approximation, but subjected to a progressive deterioration
    • Renewed ambition:
      • better foundation and structure of the field,
      • better connection with conceptual development in physics, and more globally more emphasis put on the role played by this domain in modelling activities
    • And for fulfilling this ambition:
      • an axiom for introducing the completeness of the set of real numbers, a definition for the notion of limit
      • encouraging proofs of fundamental theorems such as the Theorem of intermediate values, but not necessarily formal proofs
      • introducing the Euler method in Première S
  • 21. Implementing this curriculum: evident difficulties
    • The difficulties met by grade 10 teachers with the new statistics syllabus
    • The difficulties met by grade 11 teachers at finding their place in the TPE multidisciplinary projects
    • The difficulties met by grade 11 and 12 teachers at finding a reasonable balance between the time to be devoted to the introduction of new ideas and techniques through motivating problems and the time necessary for practicing and consolidating these ideas and techniques
    • The difficulties met by teachers at developing experimental activities in mathematics adequately supported by the use of technology as was asked to them
  • 22. The reactions to these curricular changes: the case of statistics
    • The concern of teachers suddenly asked to teach topics out of their mathematical culture
    • Questions and debates about:
      • the possibility of approaching inferential statistics without any probabilistic background,
      • the status of computer simulations, and the meaning that students could give to the experimental work supported by these simulations
      • the knowledge that could emerge from this experimental work, and the way it could be institutionalized,
      • the connection with the teaching of probability in « Première  », then « Terminale  »
    • But also an evident consensus around the social relevance of the proposed syllabus in statistics, and a strong mobilization of educational forces for supporting teachers’ adaptations
  • 23. Ten years after: a progressive and rather positive adaptation but
    • Difficulties still not solved (effects of time constraints, gap between « Seconde » and « Première S », limited mathematical autonomy given to students, marginal role given to experimental activities, modelling and real applications, limited use of technology…)
    • The increasing discrepancy between the number of students in maths, physical sciences and biology specialties resulting from the introduction of an experimental proof in sciences and biology at the Baccalauréat
    • The fact that the S orientation (above all with maths specialty) remains an « excellence » orientation more than a scientific orientation.
  • 24. The current projects
    • An attempt made for introducing an experimental proof also in maths at the baccalauréat for promoting experimental activities, the use of technology, and restoring some balance between specialties
    • An increased sensitivity to the fact that, in our educational system, the success of an elite has as a counterpart the exclusion of too many students, thus the attention paid to more inclusive systems as those from Nordic countries for organizing compulsory education (including grade 10)
  • 25. The current projects
    • A project of more flexible curriculum with a common core in grade 10 widely accessible, and a mathematical option offering insights on potentially attractive themes (models for evolution, cryptography, graph theory…), the reinforcement of connections between maths and computer sciences (an initiation to algorithmics becoming part of the common core)
    • But still fuzzy projects that the government has tried to impose without discussion for immediate application in September 2009, which has generated an intense protestation from the different communities resulting in the postponing of the reform.
  • 26. Some more general remarks
    • The complexity and fragility of educational systems
    • The necessity of adapting these to the evolution of educational needs and means, but the limits of top-down systems, where curricular changes, even if carefully thought
      • are brutally implemented
      • are not sufficiently prepared by preliminaries experimentations
      • are not enough supported by adequate training for teachers
      • are not, once introduced, carefully observed and evaluated in order to organize the necessary regulations
    • The need for more research concerning the real effects of curricular changes and allowing us to better understand the associated dynamics
  • 27. Negative strong reactions but also…
    • The progressive acknowledgement of the social relevance of the aims underlying the new syllabus
    • Joint efforts made in order to help teachers:
    • by the group of experts for producing teaching resources, and by the MEN for their dissemination
    • by the APMEP and the IREMs
    • through the organization of specific training sessions for teachers
  • 28. The situation today… No longer apparent opposition Training demand decreasing What is the exact life of Statistics in the secondary Curriculum? The necessity of research