Studying the continuity of a real function in Tunisian secondary schools: from an intuitive approach to a formal definition
Abstract
Keywords:
Teaching, Continuity of real
functions, Formal definition,
praxeology.
Continuity, as a fundamental property of real functions, occupies an important
place in the teaching of Real Analysis. However, its rigorous definition remains
problematic.
This study focuses on the didactic approach to continuity in Tunisian curricula,
by analyzing the praxeological organizations proposed in the textbook for the 3rd
high school, Mathematics section class. The analysis is based on the theoretical
framework of the Anthropological Theory of Didactics (TAD). The results show
an evolution in the order of introduction of the notions of limit and continuity,
moving from limit-continuity to continuity-limit, with a formal definition of
continuity. This reform seems to aim for greater mathematical rigor, while
relying on graphical and kinematic intuitions. However, the study of
praxeological organizations reveals difficulties in the articulation between the
different registers (intuitive, graphical, and formal) around the definition of
continuity. Some proposed activities aim to progressively construct this formal
definition, but their implementation still seems problematic. These results raise
questions about the didactic conditions that promote a satisfactory understanding
of the continuity of real functions by students, particularly the balance between
intuitive, graphical and formal approaches. Areas for improvement are proposed
for a better articulation between the different aspects in teaching.