Modelagem matemática na perspectiva da análise de modelos: indícios do pensamento matemático avançado no estudo de equações diferenciais ordinárias

Abstract

This research focuses on Mathematical Modeling from the perspective of Model Analysis and aims to investigate aspects of Advanced Mathematical Thinking (AMT) among students of a Mechanical Engineering course during Ordinary Differential Equations (ODE) classes. The research is associated with UTFPR, which has been implementing various initiatives in response to high dropout and failure rates in subjects that predominantly cover mathematical content. The research also seeks to promote the use of digital technologies in advancing Mathematical Modeling, particularly emphasizing the encouragement of Simulation in the Analysis of classical Models in Engineering courses. Furthermore, it aimed to provide Mathematical Modeling activities that facilitate the progress and observation of students' Advanced Mathematical Thinking, focusing on the Analysis of classical Models from the literature, especially those relevant to engineering courses. This research discussed the theoretical assumptions about Mathematical Modeling, Advanced Mathematical Thinking, and Model Analysis, with the use of Simulation and Digital Technologies. The research is qualitative, based on Content Analysis. The problem seeks to find evidence of students' Advanced Mathematical Thinking during the performance of Mathematical Modeling activities from the perspective of Model Analysis. The research participants are students from the 3rd period of an Engineering course, and data were collected in the context of regular and in-person classes. After the research was completed, it was observed that the adopted analytical method highlighted all the processes of AMT, revealing signs of abstraction at the end of the activities, while the other processes were identified during their execution. Similarly, in Model Analysis, it was noticed that all phases were addressed according to the development of activities. Regarding Mathematical Modeling, all pertinent steps were followed; consequently, the suggested activities achieved the proposed objectives. Thus, we conclude that an environment was created that encouraged students to adopt an investigative approach, taking the lead role in their own learning as they studied both the acceleration of a body on an Inclined Plane and the motion of a Simple Pendulum. However, it became evident that restricting analyses to a specific group reduces the observable characteristics, particularly when the activities have the potential to reveal several other evidences, thus broadening the scope of research to new areas. The Educational Product, also a result of the research, is an interactive e-book containing two Mathematical Modeling activities aimed at highlighting the students' Advanced Mathematical Thinking through Model Analysis and the practical application of these concepts. This product seeks to offer activities that allow students to apply their knowledge in problem situations while contributing to the mobilization of students' Advanced Mathematical Thinking.