Theorems or procedures? Exploring undergraduates’ methods to solve routine problems in linear algebra

Abstract

While the lion’s share of mathematics education research in linear algebra has been concerned with the conceptual aspects of students’ learning, this study focuses on procedural knowledge that undergraduates can develop in a typical first-year course. The study analyzes the methods that a large cohort of second-semester students applied in a final exam to solve three routine problems: finding a basis for a matrix column space, solving a linear system of equations, and devising a normal to a subspace. The results indicate an inverse relation between the efficiency of the methods that students used and the number of students who used them. This was especially evident in the case of theorems, the application of which was avoided by all but a few students in favor of compound algebraic procedures. Furthermore, the study shows that at the end of an instructional sequence, many students can struggle with choosing an appropriate problem-solving method and carrying it out without mistakes. The analysis of students’ mistakes is leveraged to draw university teachers’ attention to common difficulties in linear algebra.