Real numbers and Klein's double discontinuity: the intertwining between epistemological and institutional issues


Laura Branchetti, U. of Milano


The term “real numbers” is namely omnipresent in the teaching of mathematics in high school and university in Italy. However, the understanding of key properties of the field of real numbers (R) is not guaranteed at all, for students and even teachers at school and university. Potential connections between properties traditionally presented at secondary school and advanced mathematics are often not considered by teachers, who “polarize” their reflection about possible teaching strategies between intuition of the number line and decimal numbers and formal constructions of R (Branchetti 2017a,2017b). This discontinuity usually characterizes the transition from secondary school to university, where students are asked to jump from intuition to formalization of the properties of R (Bergé, 2010; 2008). In this paper, relying on an historical-epistemological analysis and a qualitative analysis of empirical data, collected during a classroom implementation carried out in a grade-12 class, the possibility of teaching real numbers through formal constructions relying on the so-called “intuitive knowledge” of the number line is problematized. This result suggests that a third way towards conceptualization of real numbers is necessary in order to allow students to grasp and manage the deep semiotic, epistemological and technical complexity behind and beyond the formal properties of the field of real numbers, that are often taken for granted and rarely fully understood by the students.


Bergé, A. (2008). The completeness property of the set of real numbers in the transition from calculus to analysis. Educational Studies in Mathematics, 67, 217-235.

Bergé, A. (2010). Students’ perceptions of the completeness property of the set of real numbers, International Journal of Mathematical Education in Science and Technology, 41(2), 217–227.

Branchetti, L. (2017a). Teaching real numbers and continuum in secondary school: an onto-semiotic approach to the investigation of teachers' choices. In: Contreras, J. M., Arteaga, P., Cañadas, G. R. , Gea, M. M., Giacomone, B., López-Martín, M. M. (Eds.). Actas del Segundo Congreso Internacional Virtual sobre el Enfoque Ontosemiótico. 

Branchetti, L. (2017b). High school teachers’ choices concerning the teaching of real numbers: a case study. In: Dooley, T., & Gueudet, G. (Eds.). (2017). Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME 10, February 1-5, 2017). Dublin, Ireland: DCU Institute of Education and ERME, 2009-2016.