Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(3) November 2009 From Informal Proceedings 29-3 (BSRLM) available at bsrlm.org.uk © the author - 85 An exploration of mathematics students’ distinguishing between function and arbitrary relation. Panagiotis Spyrou, Andonis Zagorianakos Department of Mathematics, University of Athens, Greece This paper focuses on students’ awareness of the distinction between the concepts of function and arbitrary relation. This issue is linked to the discrimination between dependent and independent variables. The research is based on data collected from a sample of students in the Department of Mathematics at the University of Athens. A number of factors were anticipated and confirmed, as follows. Firstly, student difficulties involved vague, obscure or even incorrect beliefs in the asymmetric nature of the variables involved, and the priority of the dependent variable. Secondly, there were some difficulties in distinguishing a function from an arbitrary relation. It was also thought that additional problems occur in the connotations of the Greek word for function, suggesting the need for additional research into different linguistic environments. Introduction The concept of function is essential in the understanding and learning of mathematics. It is considered to be the most important concept learnt from kindergarten to college or university (Dubinsky & Harel 1992). The difficulties students experience with this concept can only be understood in relation to its definition and the appearing of cognitive obstacles. Several researchers found that in the early stages of function teaching in secondary schools that natural models dominate using mainly 1-1 (one-to-one) functions. (Evagelidou, Spyrou, Gagatsis, & Elia 2004; Elia & Spyrou 2006). The reliance on the natural models means that the connection between the dependent and independent variable is emphasized rather than focusing on the priority of dependent to the independent variable. Furthermore, the natural models which are offered to the pupils are idealized, distant from the realities from which they were created and described in analytical formula, thus making it “difficult for the students to distinguish between relationships discovered by experience and the mathematical models of these” (Sierpinska 1992, 32). This approach results in a difficulty in realizing that the dependent variable is a magnitude which is used to estimate a measurement and that the independent variable is the means for this particular purpose, with or without an analytical formula. The etymology of the Greek word for “function” introduced a note of caution. The root of the Greek word for function (“synartisi”) is different from the origin of its Latin equivalent which is mainly operational. In colloquial Greek when a person or abstract phenomenon such as time, speed or measurement has a functional relation (“synartate”) with another person or abstraction, the effects tend to be symmetrical. A bond is implied, whether active or inert, which is triggered when “one side” (usually either side) is altered, evoking a change in the “other side”. Therefore, the common perception of the Greek word for “function” implies the symmetry of the function variables. This symmetry might create a difficulty in understanding the difference between the variables in the mathematical definition, i.e. which is the means and which is the one to measure. Sierpinska (1992) recognized this difficulty as the obstacle, “regarding the order of variables as irrelevant” (p. 38). This definition of the obstacle is the starting point of this paper.