Secondary prospective teachers grappling with ordered dense or discrete denumerable number sets: a pilot study

In this paper we present a pilot study with master students, who are also secondary prospective teachers in mathematics, where some cognitive conflicts emerged on the issue of ordered number sets being at the same time dense and denumerable. We first provide a literature review and historical-epistemological background, then we carry out an institutional overview related to these notions. Finally, we present qualitative analyses of data collected in our universities: one case study regarding a student in Italy, that made our main hypothesis emerge; a questionnaire to 30 French students. We carried out the analyses under the lenses of concept image and concept definition, which we use to analyse empirical data, collected in both countries. The analysis revealed that the concept images of discrete and dense sets in general are not connected by the students to formal definitions, and conflict emerged at the level of concept images. Since the articulation of definitions and images is crucial in the transition school- university and these concepts are basic in advanced mathematics, we discuss some implications of such conflicts and open new research questions to better investigate the problem in future studies.