Historical Conceptual Developments and The Teaching of Mathematics: From Philogenesis and Ontogenesis Theory to Classroom Practice

Abstract

More than a century ago, Hieronymus Georg Zeuthen wrote a book about the history of mathematics (Zeuthen, 1902). Of course, this was not the first book on the topic, but what made Zeuthen’s book different was that it was intended for teachers. Zeuthen proposed that the history of mathematics should be part of teachers’ general education. His humanistic orientation fitted well with the work of Cajori, 1894 who, more or less by the same time, saw in the history of mathematics an inspiring source of information for teachers. Since then, mathematics educators have increasingly made use of the history of mathematics in their lesson plans, and the spectrum of its uses has widened. For instance, the history of mathematics has been used as a powerful tool to counter teachers’ and students’ widespread perception that mathematical truths and methods have never been disputed. The biographies of several mathematicians have been a source of motivation for students. By stressing how certain mathematical theories flourished in various countries, the diverse contributions of various cultures to contemporary mathematics becomes evident. Specialized study groups have emerged in the past years as a result of the increasing interest in the history of mathematics in educational circles. Two of these are the Commission INTER-IREM Epistemologie et Histoire des Mathematiques in France and the International Study Group on the Relations between History and Pedagogy of Mathematics, which is related to International Commission on Mathematical Instruction (ICMI). In addition, regular conferences are organized, such as the European Summer Universities on the History and the Epistemology in Mathematics Education (see Lalande, Jaboeuf, & Nouaze, 1995, and Lagarto, Vieira, & Veloso, 1996, for proceedings). Concomitantly, an important numberof books are now available to help teachers use the history of mathematics (Calinger, 1996; Chabert, Barbin, Guillemot, Michel-Pajus, Borowczyk, Djebbar, & Martzloff, 1994; Dhombres, Dahan-Dalmedico, Bkouche, Houzel, & Guillemot, 1987; Fauvel & van Maanen, 2000; Katz, 2000; Reimer & Reimer, 1995; Swetz, Fauvel, Bekken, Johansson, & Katz, 1995). Instead of offering an overview of the different domains in which the pedagogical use of the history of mathematics is now ramified, we want, in this chapter, to focus on something that Cajori started and in which mathematics educators interested in the history of mathematics are still involved. That is, in considering history not only as a window from where to draw a better knowledge of the nature of mathematics but as a means to transform the teaching itself. The specificity of this pedagogical use of history is that it interweaves our knowledge of past conceptual developments with the design of classroom activities, the goal of which is to enhance the students’ development of mathematical thinking.