Conceptualization of the continuumn an educational challenge for undergraduate students

Viviane Durand-Guerrier (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck

The continuum is one of the most difficult mathematical concepts for undergraduate students. We hypothesize that among the difficulties they face in relation with this notion, the graphical evidence provided by the number line fosters the idea of a dichotomy between discreteness and continuity, hiding the property of density-in-itself, i.e. the intrinsic density with respect to order in a totally ordered set. In this paper, we first provide evidences of the weakness of fresh university students’ knowledge about real numbers. Then, we briefly present Dedekind’s construction of real numbers, which relies on the intuitive idea of the continuous line. Finally, we present a didactical situation aimed at fostering the understanding of the relationships between discreteness, density-in-itself and continuity for an ordered set of numbers.

Mots clés: The continuum; Density-in-itself; Number line; Didactical situations; Discreteness

Domaines: Histoire et perspectives sur les mathématiques [math.HO]; Education

Viviane Durand-Guerrier. Conceptualization of the continuumn an educational challenge for undergraduate students. International Journal of Research in Undergraduate Mathematics Education, 2016, 2 (3), pp.338-361. ⟨10.1007/s40753-016-0033-2⟩. ⟨hal-01413587⟩