On Some Aspects of Students’ Thinking in Linear Algebra

Abstract

This chapter1 will focus on certain aspects of students’ reasoning in linear algebra that may be responsible for their perceived difficulties in the domain. On the most general plane, it will be argued that students tend to think in practical rather than theoretical ways, and several examples will illustrate how this tendency may adversely affect their reasoning in linear algebra. On the more specific plane, three modes of reasoning in linear algebra will be distinguished, corresponding to its three interacting ‘languages’: the ‘visual geometric’ language, the ‘arithmetic’ language of vectors and matrices as lists and tables of numbers, and the ‘structural’ language of vector spaces and linear transformations. Examples will illustrate students’ reluctance to enter into the structural mode of thinking, and, first and foremost, their inability to move flexibly between the three modes. The examples will be taken 2 from a series of projects conducted at Concordia University in the years 1993-1999 .