Student connections of linear algebra concepts: an analysis of concept maps

Abstract

This article examines the connections of linear algebra concepts in a first course at the undergraduate level. The theoretical underpinnings of this study are grounded in the constructivist perspective (including social constructivism), Vernaud's theory of conceptual fields and Pirie and Kieren's model for the growth of mathematical understanding. In addition to the existing techniques for analysing concept maps, two new techniques are developed for analysing qualitative data based on student-constructed concept maps: (1) temporal clumping of concepts and (2) the use of adjacency matrices of an undirected graph representation of the concept map. Findings suggest that students may find it more difficult to make connections between concepts like eigenvalues and eigenvectors and concepts from other parts of the conceptual field such as basis and dimension. In fact, eigenvalues and eigenvectors seemed to be the most disconnected concepts within all of the students’ concept maps. In addition, the relationships between link types and certain clumps are suggested as well as directions for future study and curriculum design.

Keywords:

• concept maps
• linear algebra
• conceptual fields
• mathematical understanding
• representations

Acknowledgements

The authors are grateful to an anonymous reviewer for several helpful comments and the suggestion of additional references, all of which improved this article. We also thank Professor Dennis St John of Central Michigan University for his careful reading of the manuscript and his useful comments.