From the continuity of the line to the completeness of the field of real numbers: foundational and didactical challenges
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
Date: 1 July 2025, 13:45
Venue: 15 Norham Gardens, 15 Norham Gardens OX2 6PY
Venue Details: Seminar Room D
Speaker: Associate Professor Laura Branchetti (Department of Mathematics “Federico Enriques” Università Degli Studi di Milano, Italy)
Organising department: Department of Education
Organiser: Professor Sibel Erduran (University of Oxford)
Booking required?: Not required
Audience: Public
Editors: Hannah Freeman, Heather Sherkunov, Kristina Khoo