ABSTRACT
During the Modern Mathematics or New Math era, the introduction of number sets was one of the secondary school subjects that was experimented with. Roberts (2023) gives an example of material from the University of Illinois Committee on School Mathematics, on how integers are defined as equivalence classes of ordered pairs of natural numbers. At the Royaumont Seminar of 1959, both Jean Dieudonné and Gustave Choquet proposed to introduce the Peano axioms for natural numbers in secondary education (OEEC, 1961). From the point of view of secondary education, the previous examples can be considered as extreme proposals. With this in mind, one may wonder what alternative ways of defining number sets have found their way into secondary education during the Modern Mathematics era. Therefore, in this contribution we want to answer the research question: “To what extent have extreme proposals for the introduction of number sets actually been implemented in secondary education during Modern Mathematics?”.
As made clear in De Bock et al. (2019), Georges Papy’s Mathématique Moderne textbook series was of paramount importance in the Modern Mathematics movement, both in its country of origin, Belgium, and abroad. We therefore begin our investigation of the introduction of number sets in Modern Mathematics with a detailed discussion of the relevant content in Papy (1963, 1965). We pay particular attention to how binary numbers are used by Papy, and how Dedekind’s theorem (on the characterization of an infinite set) and Cantor’s theorem (on the uncountability of the open interval from 0 to 1, and hence of the set of real numbers) are proved for 12- and 13-year-old pupils, respectively. In addition, while the reception of Papy’s Mathématique Moderne textbook series by contemporaries is discussed in general terms in De Bock et al. (2019), in this contribution we focus in detail on how the treatment of the topic of number sets in Papy (1963, 1965) was received by contemporaries. In doing so, we shed light on a remarkable treatment of the number sets during the period of Modern Mathematics in Belgium while also paying attention to how it was perceived abroad.
SELECTED BIBLIOGRAPHY
De Bock, Dirk, Roelens, Michel & Vanpaemel, Geert (2019). Mathématique moderne: A pioneering Belgian textbook series shaping the New Math reform of the 1960s. In K. Bjarnadóttir, F. Furinghetti, J. Krüger, J. Prytz, G. Schubring & H. J. Smid (Eds.), “Dig where you stand” 5. Proceedings of the fifth International Conference on the History of Mathematics Education (pp. 129–145). Utrecht, The Netherlands: Freudenthal Institute.
OEEC (1961). New thinking in school mathematics. Paris, France: OEEC.
Papy, Georges (1963). Mathématique moderne 1. Brussels, Belgium: Didier.
Papy, Georges (1965). Mathématique moderne 2. Nombres réels et vectoriel plan. Brussels, Belgium: Didier.
Roberts, David Lindsay (2023). The rise of the American New Math movement: How national security anxiety and mathematical modernism disrupted the school curriculum. In D. De Bock (Ed.), Modern mathematics. An international movement? (pp. 13–35). Cham, Switzerland: Springer.