Difficulties in the learning and teaching of linear algebra : a personal experience

Two different personal experiences of teaching Linear Algebra are analyzed: (1) teaching a college-level linear algebra course and (2) being the instructor in a predesigned experiment that investigates a geometric approach to the concepts of vector and linear transformations using the dynamic geometry software Cabri-géomètre II . The analysis is conducted within a framework of three perspectives on students' difficulties in learning Linear Algebra: (a) the nature of Linear Algebra, (b) the didactic decisions made in teaching Linear Algebra, and (c) students' ways of thinking and their mathematical backgrounds. The historical look at the subject's development reveals that the content of an undergraduate linear algebra course is often the end product of a long process of intellectual struggle and research into deep mathematical problems with which students may never become acquainted in the course of their studies. The experiment tries to build up concepts using geometry instead of giving the final product, but fails to eliminate the structural approach. As a result, students still struggled with concepts. The detailed discussions of the situations result in an "interpretive understanding" of the situations. Recommendations on improving college-level Linear Algebra courses, such as the concentration on computations, and future research projects are given.