Learning the concept of eigenvalues and eigenvectors: a comparative analysis of achieved concept construction in linear algebra using APOS theory among students from different educational backgrounds

Abstract

International comparative studies of education have shown that the increase in mathematical attainment differs significantly among students in relation to the different school types in which their mathematical foundations were acquired. In tertiary education, the same differences are observed with respect to simple tasks that relate to the subject matter of secondary education. Therefore, we investigate the question of whether this observation holds equally for concept construction in the field of eigen theory, by applying Dubinsky’s action, process, object, schema (APOS) theory, and Tall’s worlds of mathematics. We focus on eigen theory because we regard it as particularly representative of linear algebra in terms of its expressibility in the embodied and symbolic worlds. In our empirical study, we investigated concept constructions as correlates of educational trajectories of 36 students who covered the entire course program of a first semester mathematics course for first-year engineering students. Although significant differences in procedural knowledge were observed, results indicate that the type of secondary school does not affect concept construction among students who complete the entire course program. In our final discussion, we present implications for teaching and learning of linear algebra in heterogeneous classes and the integration of deeper learning methods into the course design.