Dynamic conceptualisation of density of Q in R: Upper-secondary school students’ spontaneous conceptions and arguments

While research in Mathematics Education has mainly focused on density of ℚ, little is known about density of ℚ in ℝ, despite its importance in grasping fundamental Calculus concepts such as continuity. This study investigates upper-secondary school students’ spontaneous conceptions and arguments regarding the density of ℚ in ℝ. A classroom activity and group interviews focused on continued fractions, which provide a unified representation of rational and irrational numbers, were carried out. The data consist of the students’ written and spoken statements, and the data analysis strategy follows a thematic analysis approach with some specific modifications. The research findings indicate that students’ spontaneous conceptions are dynamic processes that range over different categories supported by different arguments and warrants.