Anna Sierpinska - Practical, theoretical, synthetic and analytic modes of thinking in linear algebra

The talk will focus on certain aspects of students' reasoning in linear algebra that may be responsible for their perceived difficulties in the domain. It will be argued that students tend to think in practical rather than theoretical ways and that, among the theoretical ways of thinking, the structural mode is the least accessible. The different modes of thinking will be described in detail, and the ways in which some of them may function as obstacles to students' understanding will be illustrated by examples of actual student behavior.

The examples will be drawn from a series of research projects conducted at Concordia University in the years 1993-1999 by myself and Joel Hillel in collaboration with, at various stages, Tommy Dreyfus and Jana Trgalova. In the years 1993-96 our research focused on patterns of interaction between individual students, their tutors and texts and their influence on the students' learning of linear algebra. The mathematical content in these observations was not designed by the researchers: it was the mathematical content of the existing linear algebra undergraduate texts. On the other hand, the design of an entry into linear algebra was at the core of the projects in the years 1996-1999. The design was based on a geometric model of the two-dimensional vector space within the dynamic geometry environment of the Cabri-geometry II software. The design had undergone a cycle of three experimentations followed by amendments to the design on the basis of the students' reactions.