O número racional na forma fracionária no ensino médio: uma abordagem conceitual, vinculada ao estudo da função polinomial de 1º grau

Abstract

This research had as main object to identify, in the conceptual approach of rational numbers in the fractional form, perspectives for the development of the operative capacity of high school students. Perceptions of the author-researcher's practice, supported by the literature, suggest that the persistence of difficulties may be associated with the prevalence of treatment based on the recollection of rules and operative procedures. Questions about the limitations of interventions based on revisions and resumptions of the theme, in High School, influenced us to link exploration of concepts and operations with rational numbers in fractional form to the study of the 1st Degree Polynomial Function. Focusing on learning, our investigation was guided by the following questions: What can be behind the difficulties in understanding some concepts and operations involving the fractional form of rational numbers, in High School? How to approach subjects in high school study to favor conceptual and operative acquisitions with rational numbers in fractional form, not yet consolidated by students? Could a conceptual approach facilitate the understanding of the rational number in fractional form and of operations with these numbers, in situations of studying the 1st degree polynomial function? The methodology involved conducting an exploratory study with 1st year high school students, which allowed us to assemble some of their doubts and tendencies in dealing with the operations in this numerical set. The process of elaborating the activities was influenced, above all, by theoretical references linked to the defense of the importance of ensuring that students conviviality, discussion, and attribution of meanings to crucial notions to the understanding of fractions and to encourage the use of mathematical ideas and knowledge of the critical and reflective mode. Considered the intention to explore situations of study of the 1st degree polynomial function with a view to favoring conceptual and operative understanding. The organization of activities in categories A, B and C allowed us to propose a situation or problem (A), intervene in students' doubts (B) and expand their contact with the relations and concepts discovered (C). The book Fractions in High School: linking the Study to Polynomial Functions of 1st Degree, educational resource produced by the research, presents the main theoretical foundations that supported our study, the set of activities produced and our discussions on perspectives and limits of the strategy used in favoring the operative capacity of high school students. We believe that this research fulfills the purpose of discussing and proposing a form of approach in which ideas associated to the concept of fraction, to the equivalence relation between fractions and to addition and subtraction operations between rational numbers in fractional form are treated as objects of study, integrated with theme of the Mathematics curriculum, with the potential to intervene in operative inabilities that persist in the final stage of Basic Education.