Teaching eigenvalues and eigenvectors using models and APOS Theory

• A modeling problem and a genetic decomposition that proved to promote the learning of eigenvalues and eigenvector are presented.
• Three students were found to construct an object conception of these concepts.
• The modeling situation proved to be useful in engaging students in the solution of the problem. It also promoted the use of previously studied concepts.
• Students discovered by themselves what later would be defined as eigenvalues and eigenvectors of a matrix, and some of their properties.
• Conceptual activities played a fundamental role in promoting deep reflection necessary to construct a process or object conception of these concepts.

In this article we share the results of an investigation of a classroom experience in which eigenvalues, eigenvectors, and eigenspaces were taught using a modeling problem and activities based on APOS (Action, Process, Object, Schema) Theory.We show how a sample of 3 students were able to construct an object conception of these difficult concepts in one semester course—something that existing literature had proven to be almost impossible to achieve. Using one team as a case study we describe the work done by a group of 30 students to show how eigenvectors and eigenvalues emerged in a group discussion. Furthermore, we present evidence on how, at least three students, were able to construct an object conception, demonstrating a deep understanding of these concepts. Finally, we validate the designed genetic decomposition. In summary, the results demonstrate the approach to be promising in the learning of eigenvalues and eigenvectors.