Teaching calculus / analysis at the turning point between Secondary School andUniversity. Knowledge, knowing and conditions for validation.

Bloch, Isabelle

HAL, NNT: 2174 / Harvested from HAL

Résumé

The study of maths curriculum in the last grades of secondary schools in France along 30years brings to light important variations that took place since 1962 in the contents of calculusat this level. These evolutions concern the objects of calculus that are taught as well as theprocedures used by students and teachers. The suggested methods affect the knowledge thatstudents are likely to use when doing the given tasks; and we observe that since the 90ths', thetasks given to students do not valorise validation. We study the possibilities of establishing anreal relationship to the knowledge in calculus, at this level of teaching, and to allow thestudents 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; howthese 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, andthe possibility of implement such situations in the class.Besides, the study of the different settings of representation that are at stake to build a suitableenvironment 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-didacticalcomponent for the teaching of function and limit, and through the observation of theirimplementation in a scientific class of 17 years-old students. This makes us first question theknowledge and professional knowing a teacher uses to manage a teaching situation inanalysis, 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 testthe main features of the learning.In the last chapter we study lectures at undergraduate level, and student’s papers with lots oferrors about calculus definitions. This leads us to question the knowledge that is compulsoryat University level; we wonder how it is possible to link it with Secondary school's knowledgeand habits.As a conclusion, we shall suggest some remarks about the balance between definitiveknowledge and what students must get as an experience in the teaching of a newmathematical theory; this balance affects the possibilities of validation and finally, the futureprospects of teaching analysis from Secondary Schools to University.

Détail

Publié le : 2000-01-19
Classification:  Calculus,  functions,  graphical representations,  limits,  validation in teaching analysis,  Theory of didactical situations and fundamental situation for Analysis,  knowledge and knowing,  ostensifs formels de fonctions,  théorie des situations et situation fondamentale de l’analyse,  savoirs et connaissances,  fonctions,  représentations graphiques,  limites,  Analyse,  [SHS.EDU]Humanities and Social Sciences/Education,  [MATH]Mathematics [math]