Promoting Linear Algebraic Reasoning among Students: Affordances and Challenges

As students transition from the mathematics they learn in school years, including their first-year calculus courses, to the first course in linear algebra, they experience discontinuities in their perspective of what mathematics is. Their propensity to continue applying the same habits of learning in the face of this change leads to failure and frustration. The failure manifests itself in the quality of understanding basic concepts as well as in the lack of linear algebraic reasoning. Instructional treatments applied in my teaching experiments to foster students' ability to reason linear algebraically resulted in mixed success – some of the treatments were successful, others less so. The latter are accounted for by the structural complexity of the subject matter and students' background knowledge. The pedagogical approaches offered in this paper are oriented within a particular theoretical framework for the learning and teaching of mathematics, called DNR. Reflections and broader implications are addressed through the lenses of this framework.