Maria Trigueros and Rafael Martinez-Planell have written an article entitled Geometrical representations in the learning of two-variable functions. The article was published online in Educational Studies in Mathematics on Wednesday. Here is the abstract of their article:

This study is part of a project concerned with the analysis of how students work with two-variable functions. This is of fundamental importance given the role of multivariable functions in mathematics and its applications. The portion of the project we report here concentrates on investigating the relationship between students’ notion of subsets of Cartesian three-dimensional space and the understanding of graphs of two-variable functions. APOS theory and Duval’s theory of semiotic representations are used as theoretical framework. Nine students, who had taken a multivariable calculus course, were interviewed. Results show that students’ understanding can be related to the structure of their schema for R³ and to their flexibility in the use of different representations.