Transitional points in constructing the preimage concept in linear algebra

Challenges related to the teaching of mathematics in higher education today

“…Au Mexique par exemple, ceci a commencé au « Departamento de Matemática Educativa del Centro de Investigación y de Estudios Avanzados -Cinvestav 9 » (département de recherche sur l'enseignement des mathématiques du Cinvestav) fondé en 1975, par Carlos Imaz-Jahnke, mathématicien reconnu, qui a étudié les modèles infinitésimaux et leur potentiel didactique pour l'enseignement du calcul, de l'analyse et des équations différentielles (e.g., Imaz, 1984). À la suite de ces travaux pionniers, des recherches se sont développées donnant lieu à des propositions didactiques alternatives (e.g., Imaz & Moreno, 2010) ; à de nouvelles approches théoriques comme la socio-épistémologie (Cantoral, 2020) ; à des études concernant l'enseignement de l'algèbre linéaire (e.g., Oktaç et al, 2021) La 2 e conférence INDRUM a eu lieu en 2018 à Kristiansand (Norvège), organisée par l'université d'Adger et le centre MatRIC. La 3 e conférence devait se tenir à Bizerte (Tunisie) en mars 2020.…”

What’s new with APOS theory? A look into levels and Totality

“…Starting from the premise that learning can be explained in terms of transitions (Gueudet et al, 2016) and considering that transitions are at the core of APOS theory to explain mental constructions of mathematical concepts, Oktaç et al (2021) investigate the understanding of the notion of preimage of a linear transformation. Considering that each transitory phase merits a study of its own, they explore the passage from an Action to a Process conception, making use of levels in APOS Theory.…”

“…(Arnon et al, 2014, p. 139). Oktaç et al (2021) report on the application of the following task (translated from Spanish) to 31 university students as part of a questionnaire, with the intention to investigate the construction of the linear transformation concept from a functional viewpoint as well as related notions such as domain, image and preimage:…”

“…In the particular case of the mentioned study (Oktaç et al, 2021), the observed levels are named as 'preimage as two vectors'; 'preimage as a set of two vectors'; ´preimage as multiple solutions' and 'preimage as infinitely many preimages'. Each of these levels corresponds to an important change in the point of view of the student when thinking about the preimage concept.…”

“…When they find the solution set as all those vectors that lie on a particular line satisfying a certain condition, they display the level 'preimage as infinitely many preimages'. Oktaç et al (2021) posit that "a level, although not as apparent as in a stage, implies a major shift in thinking about a concept". They also affirm that as transitional moments between stages, "levels evidenced in different individuals might represent different paths from one conception to another" (Figure 2).…”