The role of topology in the construction of students’ optimization schema for two-variable functions

This study uses Action-Process-Object-Schema theory (APOS) to examine students’ understanding of two-variable function optimization. A genetic decomposition (GD) based on the notion of Schema is proposed. This is a conjecture of mental structures and relations that students may construct to understand the optimization of these functions. The GD was tested with semi-structured interviews with eleven students who had just finished an introductory multivariable calculus course. Results show that giving explicit attention during instruction to the topological structure of the domain of the function to be optimized and the use of GD-based activities was effective in promoting students’ understanding of two-variable function optimization. On the theoretical side, the study contributes to a better understanding of the APOS notions of Schema, Schema-triad, and types of relations between Schema components that have not been used extensively in the literature and that proved to be a powerful tool to model students’ learning.