This talk explores the conceptual and didactical journey from the notion of continuity of the line in Euclidean geometry—embodied in the idea of the continuous line and formalized in axioms—to the formal construction of the real number system as a complete ordered field. Foundational challenges will be presented, including the key issues of the historical development of real numbers and how different constructions (Dedekind cuts, Cauchy sequences) address the notion of completeness. Moreover, the statement “real numbers are points of a line” will be problematized and analysed from a higher standpoint. On the didactical side, the talk will present a summary of the relevant literature on the topic, some open issues and the preliminary results of a study carried out in the context of a master’s course addressed to prospective secondary mathematics teachers. The goal of the course was to bridge the gap between intuition and formalism and foster a deeper understanding of the “real number line”.

Organised by Professor Sibel Erduran, Subject Pedagogy Research Group