Um estudo sobre a transição do 5º ano para o 6º ano do ensino fundamental : o caso da aprendizagem e do ensino de área e perímetro

This research looked to investigate factors of epistemological, cognitive, didactic and pedagogical natures having to do with the transition between the first and second stages of elementary education, with the area and perimeter learning objects and with its possible influence over how 6th grade students deal with these objects. The theoretical basis of this study is based on the approach of the concept area as a greatness (DOUADY; PERRIN-GLORIAN, 1989), on the conceptual field theory (VERGNAUD,1990), on the anthropological theory of the didactic (CHEVALLARD, 1999), and on the concept of recall (LARGUIER, 2009). To better comprehend the transition process, the answer to two leading questions was sought: what are the conceptual difficulties faced by students when solving situations relative to area and perimeter in the transition between elementary education’s grade 5 and grade 6? Which elements help comprehend the possible roots of these difficulties? Conducted as a case study, with a qualitative approach, the research was developed in Sao Francisco school, in the city of Recife, and included as participants students that were there in grade 5 (2016), grade 6 (2017) and grade 7 (2018); school’s directors, coordinators of both initial and final years, and mathematics teachers of both grade 5 (2016) and grade 6 (2017). To answer our questions, three studies were conducted. The first consisted of the elaboration, application and analysis of a trial test and a post-test, performed with students at the end of grade 5, and with the same students in the beginning of grade 7, respectively. The trial test was composed by 6 exercises and the post-test contained the same exercises seen in the trial test, with two additional tasks. The comparative results of the diagnostic tools show that, even after finishing grade 6, the students show difficulties related to situations that were not study objects in previous years, such as the decomposition of figures, the impossibility to tile a surface with a finite number of unitary surfaces, and the dissociations between area and perimeter. The second study consisted on the analysis of mathematics textbooks used in the school, from grade 1 to grade 6 of elementary school, of classroom observations, of students’ notes and from mathematics teachers’ planning notes. This study showed that the praxeologies taught by teachers are similar to those shown in the textbooks used, and the predominant types of tasks are to measure an area and to measure a perimeter. The third study, the comparative analysis of the grade 5 and grade 6 institutions based on the codetermination levels scale, on the official documents and on the interviews, shows internal and external pressures on the society, school and pedagogical levels, that contribute to comprehend ruptures and continuities on the transition between grade 5 and grade 6, relative to the knowledge objects area and perimeter. We observed on Sao Francisco school a phenomenon that we interpret as a paradigm conflict between the work visitation and world questioning.