Student learning of basis, span and linear independence in linear algebra

Abstract

One of the earlier, more challenging concepts in linear algebra at university is that of basis. Students are often taught procedurally how to find a basis for a subspace using matrix manipulation, but may struggle with understanding the construct of basis, making further progress harder. We believe one reason for this is because students have major difficulties with concepts of span and linear independence which form the requirements for a set of vectors to form a basis. In this research we applied a theoretical framework based on Tall's three worlds of mathematics learning and action-process-object-schema (APOS) theory to the learning of the concept of basis by a group of second year university students. The results suggest that an emphasis on matrix processes may not help students understand the concept, and embodied, visual ideas that could be valuable were usually lacking.