On the Concepts of Function and Dependence

Abstract

Abstract. This paper briefly traces the evolution of the function concept until its modern settheoretic definition, and then investigates its relationship to the pre-formal notion of variable dependence. I shall argue that the common association of pre-formal dependence with the modern function concept is misconceived, and that two different notions of dependence are actually involved in the classic and the modern viewpoints, namely effective and functional dependence. The former contains the latter, and seems to conform more to our pre-formal conception of dependence. The idea of effective dependence is further investigated in connection with the notions of function content and intensionality. Finally, the relevance of the distinction between the two kinds of dependence to mathematical practice is considered.

Grade 10 teachers’ example selection, sequencing and variation during functions lessons

“…In addition to the above discussion, for Sierpinska (1992), functions should be viewed 'as tools of description and prediction' (p. 32) of how variables are related to each other, making functions models of patterns in real-life phenomena. This resonate with Euler's and Dirichlet's definitions of what a function is: that functional relationships can be expressed in terms of covariation or by using a rule of correspondence between variables (Bazzoni, 2015;Blanton, 2008;Wilkie, 2020). In describing covariational relationships, Borba and Confrey (1996) stated:…”

“…' (p. 323) This is not sufficient to constitute a function. According to Bazzoni (2015), learners need to be taught that the association between the two variables can be understood as fixed points on a Cartesian plane and are usually represented by a set of ordered pairs as coined by Bourbaki in the form (x; f(x)). Of importance to note is that the enablement of learners' understanding of functional relationships depends on the quality of the examples that a teacher selects and uses to guide learners towards generality about specific functions concepts (Essien, 2021).…”