A noção de infinitésimo no esboço de curvas no ensino médio: por uma abordagem de interpretação global de propriedades figurais

Abstract

Sketching and understanding curves are activities that permeate several actions and areas of human life. Such activities allow the representation of everyday facts and situations, being present in elementary, middle and higher education, in different subjects. However, the educational scenario and research in Mathematics show the difficulties students, from all levels of education, have in delineating curves and in understanding the phenomenon that a curve represents. In order to problematize the understanding of curves of some functions in secondary education, we introduce an alternative path and analyze its possibilities and limitations, related to teaching and learning, in light of the global interpretation approach of the figurative properties of the semiotic registers of representation theory, recommended by Raymond Duval. The global interpretation approach is a result of studies carried out by the author with the linear/affine function, in order to analyze the congruence between the algebraic and graphical registers that permeate the discrimination of the significant units of each record and the implicit transformations required for the change of record. Duval performed his analyzes taking into account the parameters of the algebraic register of the affine function and the effects of their variation, in the graphic record. In the case of second and third degrees polynomial functions, analyzing the parameters variations becomes impracticable due to the amount of possible variation. Therefore, in order to maintain the commitment to the global interpretation, we use as a resource, the variation rates of the function comprised from the notion of infinitesimal, without the formalization and accuracy required in the work with limits concept. We traced this alternative path of sketching curves in the classroom with third year of high school students, from the elaboration and development of a didactic sequence, organized according to elements of Didactic Engineering. The reflections have allowed us to conclude that the work in this perspective allows the recognition of basic symbolic units, related to variability and basic graphic units and, more than that, the circulation between them, without the need to obtain the algebraic expression of the function. We understand, from the application of the didactic sequence and the analysis of the dialogues as well as the students resolutions regarding the organized functions of the speech, the process of knowledge appropriation by the students and the intellectual gestures mobilized in the alternative way that made possible the wide comprehension of the concept of functions in terms of their variability. In relation to teaching, we show limitations from teaching curves drawing permeated by a point-to-point approach, which limits the students' observation on the movement, transformation and dynamism present in the concept of functions, not providing the consciousness of the conversions among necessary semiotic registers, nor of the treatments. In this context, we suggest that, in secondary education, the teaching of functions and the sketching of curves should be promoted in harmony with the students' learning possibilities, allowing a differentiated look at a curve and what it represents and, more than that, a teaching that stimulates the written and oral communication of the conclusions in the resolution of mathematical problems.