Linear algebra students’ modes of reasoning: Geometric representations

Main goal of our research was to document differences on the types of modes linear algebra students displayed in their responses to the questions of linear independence from two different assignments. In this paper, modes from the second assignment are discussed in detail. Second assignment was administered with the support of graphical representations through an interactive web-module. Additionally, for comparison purposes, we briefly talk about the modes from the first assignment. First assignment was administered with the support of computational devices such as calculators providing the row reduced echelon form (rref) of matrices. Sierpinska’s framework on thinking modes (2000) was considered while qualitatively documenting the aspects of 45 matrix algebra students’ modes of reasoning. Our analysis revealed 17 categories of the modes of reasoning for the second assignment, and 15 categories for the first assignment. In conclusion, the findings of our analysis support the view of the geometric representations not replacing one’s arithmetic or algebraic modes but encouraging students to utilize multiple modes in their reasoning. Specifically, geometric representations in the presence of algebraic and arithmetic modes appear to help learners begin to consider the diverse representational aspects of a concept flexibly.