Students' understanding of the 2D Heat Equation: An APOS approach

TL;DR

This study applies the APOS framework to assess and improve students' conceptual understanding of the 2D heat equation, focusing on mental constructions related to the Laplacian and other key concepts.

Contribution

It validates a hypothetical learning trajectory for the 2D heat equation using interviews, highlighting specific mental constructions students develop and areas needing refinement.

Findings

Students successfully engage with key mental constructions of the Laplacian.

Coordination and encapsulation of process conceptions improve understanding.

Some mental constructions related to temperature distribution and heat flow require further development.

Abstract

In this paper, we use the APOS theoretical framework to validate a hypothetical learning trajectory of the 2D heat equation, a preliminary genetic decomposition that stresses the conceptual understanding of its mathematical formulation. We design questions to probe specific mental constructions of the preliminary genetic decomposition. We interview 8 students in the second year of the B.Sc. enrolled in either engineering, physics or twin (mathematics and physics) majors. Our findings indicate that students engage with many predicted mental constructions. In particular, coordination and encapsulation of two process conceptions of the Laplacian of the temperature improve understanding although it is challenging. Other parts of the genetic decomposition require refinements. These include mental constructions related to the temperature distribution function, heat flow, and the temperature gradient.