INNOVACIÓN EN LA ENSEÑANZA DE LA TRIGONOMETRÍA UNIVERSITARIA: DISEÑO E IMPLEMENTACIÓN DE UNA TRAYECTORIA HIPOTÉTICA DE APRENDIZAJE PARA ECUACIONES TRIGONOMÉTRICAS CON INFINITAS SOLUCIONES

Abstract

This doctoral thesis aimed to design, implement, and evaluate a hypothetical learning trajectory (HLT) to facilitate the construction of the concept of trigonometric equations with infinite solutions among first-year engineering students. The theoretical framework was based on the error typologies proposed by Radatz (1979), Movshovitz-Hadar et al. (1987), and Socas (1997), as well as on Simon’s (1995) construct of HLT and the design heuristic of emergent models (Gravemeijer, 1999). The methodology followed was design-based research._x000D_ The study was carried out in three phases. In the first phase, a study was conducted on the errors made by first-year engineering students when solving trigonometric equations. This study served as a key input for the design of the initial HLT for constructing trigonometric equations with infinite solutions. In the second phase, two experimental cycles were carried out with engineering students, allowing for the progressive implementation and refinement of the didactic proposal. In the third phase, a retrospective analysis was conducted on students’ progress in constructing the mathematical concept, which enabled the validation and refinement of the HLT._x000D_ The results provide evidence that the HLT supported students in progressing from a model-of the solution set of trigonometric equations with finite solutions within bounded intervals to a model-for the solution set of trigonometric equations with infinite solutions. The levels of activity in the design heuristic of emergent models offered students opportunities to build the concept of trigonometric equations with infinite solutions, starting from their informal mathematical activity (locating angles on the unit circle) toward more formal mathematical reasoning (solving trigonometric equations in context)._x000D_ This thesis offers both practical and theoretical contributions to university-level mathematics education. From a theoretical perspective, it advances didactical knowledge in university trigonometry by providing empirical evidence of the applicability of the emergent models heuristic in higher education contexts. From a practical standpoint, the systematic identification and classification of specific errors provide valuable information for curriculum design and instructional planning, while the developed HLT serves as a pedagogical tool directly applicable in geometry courses for engineering students.

Date of Award	19 Nov 2025
Original language	Spanish
Awarding Institution	Universitat Autònoma de Barcelona (UAB)
Supervisor	Josep Maria Fortuny Aymemi (Director) & Andrea Dorila Cárcamo Bahamonde (Director)