Examining university students' multivariational reasoning on two-variable functions

The aim of this study is to examine university students' multivariational reasoning processes regarding two-variable functions in a technology-enhanced collaborative learning environment and to identify the mental actions they employ during this process. The study was conducted with 15 pre-service mathematics teachers enrolled in a mathematics education program at a public university in Turkey. During the data collection process, students' audio and video recordings, computer screenshots, GeoGebra files and tasks were used. The analysis results revealed that the technology-supported collaborative learning environment, designed using the ACODESA method and GeoGebra software, supported students' processes of constructing graphs of two-variable functions, mathematically interpreting level curves, and analyzing the intersection of a plane with a function surface in three-dimensional space. In doing so, it encouraged their multivariational reasoning. It was found that students used all of the mental actions included in the multivariational reasoning framework. However, they most frequently employed the action of reduce into isolated covariations, while the action of continuous multivariation was used the least. Additionally, students interpreted relationships between variables as independent, dependent, and nested multivariations depending on the context. This finding indicates that their reasoning processes were influenced by the structural features provided by the context and that their conceptualizations were context-sensitive. The GeoGebra software used in the study enabled students to transition between multiple representations and to visualize and analyze variable relationships. It was particularly effective in concretizing abstract concepts in three-dimensional space. Accordingly, it is recommended to use technology-supported collaborative learning environments in the teaching of abstract concepts such as two-variable functions. In particular, it is important to incorporate instructional strategies that promote less frequently used mental actions, such as continuous multivariation.