Student thinking modes the case of solution sets and linear independence from first year linear algebra

Symbolic mathematics appearing in theorems and formal definitions is relatively new for the human mind. This is one reason why modes used while reasoning mathematically became the focus for many mathematicians and educators. This research was conducted using two computer systems designed to provide graphical representations of consistent and inconsistent systems of equations, and to provide algebraic representations of linear independence or dependence of vectors. The primary purpose of the thesis was to document thinking modes (Sierpinska, 2000) through classifying students' responses to linear algebra questions from two separate assignments. Without a doubt, the computer systems can serve as a linking mechanism between various representations of basic linear algebra concepts. We observed multiple modes displayed on student responses to a set of questions on the concepts of solution set and linear independence.