Students' understanding of the general notion of a function of two variables

In this study we analyze students' understanding of two-variable function; in particular we consider their understanding of domain, possible arbitrary nature of function assignment, uniqueness of function image, and range. We use APOS theory and semiotic representation theory as a theoretical framework to analyze data obtained from interviews with thirteen students who had taken a multivariable calculus course. Results show that few students were able to construct an object conception of function of two variables. Most students showed difficulties finding domains of functions, in particular, when they were restricted to a specific region in the xy plane. They also showed that they had not fully coordinated their R³, set, and function of one variable schemata. We conclude from the analysis that many of the interviewed students' notion of function can be considered as pre-Bourbaki.