The history of the concept of function and some educational implications

Abstract

Several fields of mathematics deal directly or indirectly with functions: mathematical analysis considers functions of one, two, or n variables, studying their properties as well as those of their derivatives; the theories of differential and integral equations aim at solving equations in which the unknowns are functions; functional analysis works with spaces made up of functions; and numerical analysis studies the processes of controlling the errors in the evaluation of all different kinds of functions. Other fields of mathematics deal with concepts that constitute generalizations or outgrowths of the notion of function; for example, algebra considers operations and relations, and mathematical logic studies recursive functions. It has long been argued that functions should constitute a fundamental concept in secondary school mathematics (Klein, 1908/1945) and the most recent curriculum orientations clearly emphasize the importance of functions (National Council of Teachers of Mathematics, 1989). Depending on the dominant mathematical viewpoint, the notion of function can be regarded in a number of different ways, each with different educational implications. This paper reviews some of the more salient aspects of the history of the concept of function,1 looks at its relationship with other sciences, and discusses its use in the study of real world situations. Finally, the problem of a didactical approach is considered, giving special attention to the nature of the working concept underlying the activities of students and the role of different forms of representation.