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L'enseignement de l'analyse à la charnière lycée / université
Savoirs, connaissances et conditions relatives à la validation

Abstract : The study of maths curriculum in the last grades of secondary schools in France along 30 years brings to light important variations that took place since 1962 in the contents of calculus at this level. These evolutions concern the objects of calculus that are taught as well as the procedures used by students and teachers. The suggested methods affect the knowledge that students are likely to use when doing the given tasks; and we observe that since the 90ths', the tasks given to students do not valorise validation. We study the possibilities of establishing an real relationship to the knowledge in calculus, at this level of teaching, and to allow the students to build appropriate methods.
We study the question of validation in teaching analysis through the following directions:
- the mathematical theory; its organisation; the methods of proof and the formalization; how these methods can be introduced in the teaching, in a way that students can understand;
- the existence of fundamental situations concerning the concepts of function and limit, and the possibility of implement such situations in the class.

Besides, the study of the different settings of representation that are at stake to build a suitable environment for the teaching of function and limit makes new potentialities come to light, particularly in the graphic and formal settings.
The experimentation is carried through the building of situations with an a-didactical component for the teaching of function and limit, and through the observation of their implementation in a scientific class of 17 years-old students. This makes us first question the knowledge and professional knowing a teacher uses to manage a teaching situation in analysis, with an a-didactical component, and then draw a pattern to the teacher's milieu.
We also submit a test to the students and analyse the results with statistic tools so as to test the main features of the learning.
In the last chapter we study lectures at undergraduate level, and student's papers with lots of errors about calculus definitions. This leads us to question the knowledge that is compulsory at University level; we wonder how it is possible to link it with Secondary school's knowledge and habits.
As a conclusion, we shall suggest some remarks about the balance between definitive knowledge and what students must get as an experience in the teaching of a new mathematical theory; this balance affects the possibilities of validation and finally, the future prospects of teaching analysis from Secondary Schools to University.
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Submitted on : Monday, April 17, 2006 - 11:22:04 PM
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Isabelle Bloch. L'enseignement de l'analyse à la charnière lycée / université
Savoirs, connaissances et conditions relatives à la validation. Mathématiques [math]. Université Sciences et Technologies - Bordeaux I, 2000. Français. ⟨tel-00012151⟩

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