Equações diferenciais ordinárias no contexto dos registros de representação semiótica e da modelagem matemática

Abstract

The ordinary differential equations (ODEs) have wide application in the resolution of complex problems of movement, growth, electricity, vibrations and of various types of physical phenomena which involve rates of variation of quantities in the most diverse areas of knowledge. In this context, studying these equations allows the establishment of several relations between Sciences and Mathematics. However, what is observed in the disciplines in which this subject is approached is that it is treated with a strictly algebraic approach, causing the students to worry only about learning the method that can solve it, leaving aside the main objective which would be understanding the process that generated such an ordinary differential equation (ODE), and what the relationship between the solution and the phenomenon it describes. Mathematical Modeling, one of the trends in Mathematics Education, makes it possible to establish several relationships between the mathematical model, in this case an ODE, and the analyzed phenomenon, also favoring the attribution of meanings to the variables that describe such phenomenon. Based on these assumptions, we have developed a qualitative study, aiming to investigate the potential of a sequence of situations, involving problems in the context of Mathematical Modeling, from the perspective of semiotic representation registers and interplay between settings, in the conduction of the learning process of the ODEs for Engineering students. As methodological procedures, we have proposed the development of a sequence of situations based on the presuppositions of didactic engineering. The results showed that the sequence of situations enabled the students to understand the ODEs that contemplate the knowledge pointed out in the exercises proposed in the textbooks. The study in different domains and the use of different registers of semiotic representation aided in the establishment of relations between the family of solutions and the derivative between ODE and its field of vectors and between the signal of the derivative and the field of vectors. In particular, in the development of Mathematical Modeling problems, this knowledge has allowed the students to interpret the relations between the equation and the modeled phenomenon, since this type of activity demands students to interpret the studied situation and not only solve the ODE. In this process of analysis and interpretation of the phenomenon it becomes necessary to use different tools of mathematical domains and requires the use of different representation registers, which enables the treatment, conversion and coordination between registers.