Desarrollo de un análisis didáctico-matemático conjunto. Un estudio de los factores de cambio en las creencias y concepciones de profesores universitarios sobre la enseñanza y aprendizaje del espacio vectorial

Abstract

Linear Algebra occupies a foundational position in the education of mathematicians, engineers, and scientists, serving as both a core theoretical framework and a gateway to abstract reasoning in the mathematical sciences. Yet, despite its centrality, its teaching and learning continue to pose significant challenges. The concept of the vector space represents a particularly critical juncture, demanding that learners shift from computational reasoning toward structural and relational thinking. Extensive research indicates that these difficulties are not merely epistemic or cognitive, but are deeply embedded in faculty beliefs, conceptions, and didactic–mathematical knowledge, which collectively shape instructional practices and influence how students construct meaning. This dissertation investigates university professors’ beliefs about the teaching and learning of the vector space, and the factors that contribute to or constrain their transformation. It is grounded in the premise that teachers’ beliefs are dynamic, evolving constructs that emerge from the interplay between mathematical understanding, pedagogical experience, and institutional context. Recognizing this complexity is essential to fostering a more reflective, research-informed approach to mathematics teacher education at the university level. Employing a qualitative, descriptive–interpretive research design, the study explores how professors conceptualize the mathematical objects associated with the vector space, how these conceptions are enacted in their pedagogical practices, and how reflective engagement may facilitate belief transformation. A research–training laboratory was implemented to document participants’ initial beliefs, observe the evolution of their thinking throughout the process, and identify key factors underlying change. Analytical categories were derived from a theoretical framework integrating the constructs of belief, conception, and didactic–mathematical knowledge, providing a coherent lens for interpretation. The findings reveal that meaningful transformation in teachers’ beliefs does not arise from content exposure alone but from the nature of the formative experience—specifically, the dialogic, reflective, and analytical quality of professional interactions that enable educators to reexamine the epistemological and pedagogical foundations of their practice. The study corroborates international evidence highlighting the role of didactic–mathematical analysis in cultivating teachers’ professional vision, deepening their understanding of mathematical structures, and enhancing their capacity for pedagogical reasoning.