Key note presentation, annual conference of the Finnish Association for Research in Mathematics and Science Education, Oct. 2010. To appear in proceedings Anthropological theory of didactic phenomena: some examples and principles of its use in the study of mathematics education Carl Winsløw 1 winslow@ind.ku.dk 1 Department of Science Education, University of Copenhagen, Denmark Abstract : The founder of the anthropological theory of the didactical, Yves Chevallard, was recently awarded the Hans Freudenthal medal, given in recognition of “a major cumulative program of research” in mathematics education. The aim of this paper is to present, to a Nordic audience, an outline of this theoretical programme, and to highlight some of the features which we believe to be of particular relevance to research on mathematics education. We begin by a more general discussion of the meaning of the term “theory”. Then we present, in outline, the main elements of the anthropological theory, along with some examples of the uses and contributions the author has been involved in. Based on these examples, we conclude that a main feature of the theory is to enable a precise analysis of mathematical activity as it occurs within teaching situations, in strict coherence with an analysis of conditions and constraints at “higher institutional levels” which contribute to determine this activity. Key Words: anthropological theory, didactics, mathematical education 1. Introduction “Men die, systems last.” As Hans Freudenthal completed his Didactical phenomenology of mathematical structures (1983), he left these words in epitaph of the preface. He does so with a certain ironic distance, as he qualifies this capital book as “the most chaotic” of all his works. In the book, Freudenthal develops his basic idea: that “mathematical concepts, structures, ideas, have been invented as tools to organise the phenomena of the physical, social and mental world” (ibid., p. ix). He develops this thesis, not as a theoretical construction with sparse examples to “illustrate it”, but through detailed and rich studies of concrete and fundamental mathematical themes – some of which may, indeed, appear also a