Representações semióticas mobilizadas no estudo da área do círculo no ensino fundamental
This research aims to analyze the registers of semiotic representation and the sequential, perceptual, discursive and operative apprehensions mobilized by students in 9th grade of Lower Secondary Education in the study of the area of the circle. Therefore, a qualitative research was carried out (LÜDKE; ANDRÉ, 1986) in a Municipal School of Primary and Lower Secondary Education of Erechim, in the state of Rio Grande do Sul, Brazil. The data was collected through two collections of textbooks (TB) adopted by the School in the academic year of 2016 – one destined to Primary Education, denominated Novo Girassol: saberes e fazeres do campo (Collection A), and another, to Lower Secondary Education, entitled Vontade de Saber Matemática (Collection B) –; notebooks of two students from each of the nine classes, from the 1st to the 9th grade, of the School; and a sequence of activities developed with the students in 9th grade at the School, consisting of 29 items about obtaining the area of the circle using the method of exhaustion, through manipulatives and the GeoGebra software. With the intention of establishing inferences about the study of the area of the circle in Primary and Lower Secondary Education, based on the principles of content analysis (BARDIN, 2016), it was found that the area of the circle was a subject discussed only in the 9th-grade TB, while, in the previous years, especially in Primary Education, the characteristics of the circle were explored, as well as its differences from other flat forms. The activities identified involved perceptual (72.55%), discursive (1.96%) and operative (37.25%) apprehensions, with mobilization of natural language register (NLR) and figural register (FgR) in the statements. Through the analysis of the notebooks, it was detected that the study of the area of the circle was developed in the 8th and 9th grades, mobilizing only the perceptual and operative apprehensions in exercises that were repeated in these two school years. The perceptual apprehension was employed in all activities, while the operative apprehension (50%), only to determine the area of the circular crown. With respect to the representations, it was found that, in these two data sources, except for one activity, the NLR was used only in the statements, and the approaches emphasized algebraic (AlR) and numerical (NmR) registers. The sequence of activities involved the reconfiguration of the area of the circle in the approximate form of a rectangle and the inclusion of polygons in the circle in order to manipulate different registers of semiotic representation, articulating, in particular, FgR and AlR. Still concerning the sequence, it was concluded that all apprehensions were mobilized, maintaining the prevalence of perceptual apprehension (95.24%), but with an expressive increase of discursive (34.52%) and operative (32.14%) apprehensions. In addition, the simultaneous mobilization of more than one apprehension was observed in several answers of the protocols (53.57%), mainly since they used NLR, accompanied concomitantly by more than one representation register, with emphasis on FgR, AlR and NmR, which characterizes the conversion.