Morris Orzech - Addressing student difficulties specific to linear algebra

At my university, first year students who plan a mathematics concentration as part of their degree programme take courses in linear algebra and in calculus. Each course offers its challenges, but linear algebra seems a more uncomfortable experience for both teachers and students. My outlook on this phenomenon is that linear algebra seems to be more difficult for students despite it being taught with as much care, and as much attention to content, as is calculus. This viewpoint has led me to shift my attention from an almost exclusive focus on choosing material and presenting it clearly to an increased concern with aligning the way material is offered to the students' abilities to learn it. Some of my attempts in this direction depart from my traditional teaching practice by looking for guidance to learning-theory based ideas of people like Harel, Hillel and Sierpinska. Some are more ad-hoc, but are unconventional in other ways: in the name of having students better learn and appreciate mathematics they challenge some shibboleths that our conventional teaching practice seeks to inculcate.