Action-Process-Object-Schema Theory Research Articles

Students' understanding of quadratic equations

Action–Process–Object–Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help students achieve an understanding of quadratic equations with improved interrelation of ideas and more flexible application of solution methods. Semi-structured interviews with eight beginning undergraduate students explored which of the mental constructions conjectured in the genetic decomposition students could do, and which they had difficulty doing. Two of the mental constructions that form part of the genetic decomposition are highlighted and corresponding further data were obtained from the written work of 121 undergraduate science and engineering students taking a multivariable calculus course. The results suggest the importance of explicitly considering these two highlighted mental constructions.

Eigenvalues and eigenvectors: embodied, symbolic and formal thinking

Many beginning university students struggle with the new approaches to mathematics that they find in their courses due to a shift in presentation of mathematical ideas, from a procedural approach to concept definitions and deductive derivations, and ideas building upon each other in quick succession. This paper highlights this struggle by considering some conceptual processes and difficulties students find in learning about eigenvalues and eigenvectors. We use the theoretical framework of Tall’s three worlds of mathematical thinking, along with perspectives from Dubinsky’s APOS (action, process, object, schema) theory and Thomas’s representational versatility. The results of the study describe thinking about these concepts by several groups of first- and second-year university students. In particular the obstacles they faced, and the emerging links some were constructing between parts of their concept images formed from the embodied, symbolic, and formal worlds are presented. We also identify some fundamental problems with student understanding of the definition of eigenvectors that lead to implementation problems, and some of the concepts underlying such difficulties.