The learning and teaching of linear algebra: Observations and generalizations

Abstract

This paper is about a teaching experiment (TE) with inservice secondary teachers (hereafter “participants”) in the theory of systems of linear equations. The TE was oriented within particular social and intellectual climates, and its design and implementation took into consideration a series of findings concerning the difficulties students have in linear algebra. The questions we set for this study were: (1) Did the participants in the particular TE climates construct viable knowledge in the theory of systems of linear equations? Our criteria for viable knowledge consist in evidence for the ability to (a) generate non-trivial conjectures, judged so subjectively by a mathematician, (b) prove such conjecture, and (c) move upward along the APOS conception levels. (2) What difficulties and insights did the participants experience as they constructed such knowledge?The potential contributions of our investigation into these questions to researchers and practitioners include (a) a detailed depiction of the participants’ achievements and challenges in dealing with theoretical questions concerning linear systems in an authentic learning environment and under a tutelage oriented in a particular constructivist perspective; and (b) a field-based hypothesis about the consequences of a particular learning environment vis-à-vis construction of knowledge in linear algebra.All of the participants had taken a linear algebra course as part of their undergraduate studies, on average 17 years prior to the TE, with an average grade of about 80%. Thus, a third question set for this study concerns retention. (3) What did the participants retain from their linear algebra courses vis-à-vis concepts, ideas, and problem solving pertaining to the theory of systems of linear equations, assuming they had constructed such knowledge during these courses?