Infinite Series from History to Mathematics Education

In this paper an example from the history of mathematics is presented and its educational utility is investigated, with reference to pupils aged 16-18 years. Students’ behaviour is examined: we conclude that historical examples are useful in order to improve teaching of infinite series; however their effectiveness must be verified by the teacher using experimental methods, and the primary importance of the cultural context must be taken into account. Introduction Several authors have shown that the history of mathematics can be widely employed by teachers in the presentation of many mathematical topics (Fauvel & van Maanen, 2000); of course this requires some epistemological assumptions: teaching is influenced by teachers’ conceptions about the nature and the evolution of scientific knowledge (Moreno & Waldegg, 1993; Heiede, 1996). A. Sfard states that, in order to speak of mathematical objects, it is necessary to consider the whole process of concept formation and she supposes that an operational conception can be considered before a structural one (Sfard, 1991). As regards the savoir savant (Chevallard, 1985) the historical development of many mathematical notions can be considered as a sequence of stages: an early intuitive stage and so on, until the mature stage is reached. This savoir cannot be considered absolute and it must be understood in terms of cultural institutions (Lizcano, 1993; Grugnetti & Rogers, 2000; Furinghetti & Radford, 2002). The sociological perspective is clearly relevant to mathematics education. Let us consider the approach introduced by R. Cantoral and R.M. Farfán (Cantoral, 2001; Cantoral & Farfán, 2003): while epistemological approaches sometimes do not take into account the influence of social contexts, in the theoretical approach denominated “Socioepistemology” an extension of theory of didactic situations is proposed in order to show the social construction of the knowledge and the negotiation of meanings. With reference to the use of the history into didactics on mathematics, several theoretical frameworks can be mentioned in order to link learning processes with historical issues (Cantoral & Farfán, 2004, see in particular the Chapter 8). According to the “epistemological obstacles” perspective (Brousseau, 1983), a goal of historical study is finding systems of constraints (situations fondamentales) that must be studied in order to understand existing knowledge, whose discovery is connected to their solution (Radford, Boero & Vasco 2000, p. 163). It seems that this perspective is characterised by an epistemological assumption: the reappearance in teaching-learning processes of the obstacles encountered by mathematicians in the past. Nevertheless, historical data must be considered nowadays and several issues are connected with their interpretation, again based upon our cultural institutions and beliefs (Gadamer, 1975); according to Radford’s socio-cultural perspective, knowledge is linked to activities of individuals and this is strictly related to cultural institutions; knowledge is not built individually, but in a wider social context (Radford, Boero & Vasco, 2000). In this work we shall discuss the introduction of infinite series by using some well known historical examples (concerning the history of the Calculus see: Edwards, 1994; Hairer & Wanner, 1996). When we introduce infinite series, we must keep in mind that a sum of infinitely many addends is frequently considered by pupils as “infinitely great” (Bagni, 2000a) so first of all we must overcome the misconception “infinitely many addends, infinitely great sum”. Of course it is possible to employ several visual representations (see the picture: the big square, whose side’s length is 1, is divided into a sequence of triangles so that it is possible to state: = + + + + + + ... 64 1 32 1 16 1 8 1 4 1 2 1 1. But the educational use of visual registers can cause problems, particularly if pupils’ ability to coordinate representation registers is lacking: Duval, 1995; D’Amore, 2001). The history of mathematics can help us to direct our pupils correctly; for instance, we can mention Zeno of Elea (490-430 BC) and his famous paradox of Achilles and of the Turtle: it is well known that it lead us to consider a convergent geometric series. This example can be very useful and through this, implicitly, we present a sum of infinitely many addends that cannot exceed a finite number. Unfortunately some misunderstandings can arise: in particular, pupils could notice that addends, in that case, are indefinitely small and this condition can be considered wrongly as a sufficient one for the convergence of an infinite series (the harmonic series, studied in the 14 century by Nicole Oresme is useful for overcoming this mistake: Anglin, 1994, p. 134). We shall present some historical examples that can be employed in classroom practice; then we shall examine students’ reactions in a brief experimental survey. Infinite series: historical remarks Ut non-finitam Seriem finita cöercet, Summula, & in ullo limite limes adest: Sic modico immensi vestigia Numinis haerent Corpore, & angusto limite limes abest. Cernere in immenso parvum, dic, quanta voluptas! In parvo immensum cernere, quanta, Deum! Jakob Bernoulli (Ars Conjectandi, 1713) A first notion of infinite series may well have a very ancient source: Aristotle himself implicitly underlined that the sum of a series of infinitely many addends (potentially considered) can be a finite quantity (Physics, III, VI, 206 b, 1-33). In his Quadratura parabolæ, Archimedes considered (implicitly, once again) a geometric series. Several centuries later, Andreas Tacquet (1612-1660) noticed that the passage from a “finite progression” to an infinite series would be “immediate” (Loria, 1929-1933, p. 517); but such a passage is crucial, from the epistemological point of view. In fact, Tacquet made reference to ancient mathematics without any historical contextualization: Greek conceptions strictly distinguished actual and potential infinity (mathematical infinity, following Aristotle, was accepted only in a potential sense so it is meaningless to suppose any explicit consideration of infinite series. Tacquet’s position, too, must be contextualised: we cannot suppose the presence of our epistemological awareness in the 17 century; it is necessary to take into account either the period in which the original work was written or the period of its edition or comment: Barbin, 1994; Dauben & Scriba, 2002). In particular, we are going to examine a well-known indeterminate series. In 1703, Guido Grandi (1671-1742) noticed that from the infinite series 1−1+1− 1+... it is possible to obtain 0 or 1: (1−1)+(1−1)+(1−1)+(1−1)+... = 0+0+0+0+... = 0 1+(−1+1)+(−1+1)+(−1+1)+... = 1+0+0+0+... = 1 The sum of the alternating series = 1−1+1−1+... was considered 1⁄2 by Grandi (and by several mathematicians in the 18 century). According to him, the proof can be based upon the following expansion expressed using modern notation (nowadays accepted if and only if |x|<1):