Normat – Nordisk Matematisk Tidsskrift 53, 4 (2005), 173-185 MATHEMATICS EDUCATION AND HISTORICAL REFERENCES: GUIDO GRANDI’S INFINITE SERIES Giorgio T. Bagni Department of Mathematics and Computer Science University of Udine (Italy) Abstract. Integrating history of mathematics into the mathematics education is an important point: the use of history in education can be an effective tool for the teacher. In this paper an example from the history of mathematics (Grandi’s infinite series, 1703, and Leibniz remarks, 1716) is presented and its educational utility is investigated. Students’ behaviour is examined, with reference to pupils aged 16-18 years: we conclude that historical examples are useful in order to improve teaching of infinite series, e.g. to stimulate reflections about actual and potential infinity. Nevertheless the main problem of the passage from finite to infinite is a cultural one, and historical issues are important in order to approach it, although the historical approach can be considered together with other educational approaches. Key words: actual and potential infinity, didactical contract, history of mathematics, infinite series, probabilistic argument, socio-cultural context. 1. HISTORY AND MATHEMATICS EDUCATION “Even 500 years ago a philosophy of mathematics was possible, a philosophy of what mathematics was then”. Ludwig Wittgenstein (1956, IV, 53) Several theoretical frameworks can be mentioned in order to link learning processes with historical issues (Fauvel & van Maanen, 2000; Cantoral & Farfán, 2004, see in particular the Chapter 8). According to the “epistemological obstacles” perspective (Bachelard, 1938; Brousseau, 1983), one of the main goals of historical study is finding systems of constraints (the so-called situations fondamentales) that must be studied in order to understand knowledge, whose discovery is connected to their solution. Some Authors (Radford, Boero & Vasco 2000, p. 163) notice that this perspective is characterised by an important assumption: the reappearance in our teaching-learning processes, in the present, of the obstacles encountered by mathematicians in the past. Nevertheless, historical data must be considered nowadays and several issues are connected with their interpretation, based upon our cultural institutions and beliefs; according to Luis Radford’s socio-cultural perspective, knowledge is linked to activities of individuals and this is essentially related to cultural institutions (Radford, 1997); knowledge is not built individually, but in a wider social context (Radford, Boero &