Análise dos efeitos didáticos emergentes de uma sequência de atividades na aprendizagem do significado parte/todo do número racional

Abstract

This present research is inserted into the field of Mathematics Education and is aimed at investigating the didactical effects emerging from a sequence of activities that explore the notion of fractions related to the whole/part meaning of the rational numbers. The didactical sequence, consisting of three groups of activities, was extracted from a Mathematics textbook used by teachers from a municipal education network in the metropolitan area of Recife. At one of the schools run by the referred-to network, we proposed the application of the didactical sequence and we counted on the collaboration of forty-two elementary school fifth-graders who were motivated to solve the activities in two different situations: a didactical, or learning one, and an evaluating one. In order to make the analysis of data possible we went for some theoretical foundations present in the “Theory of Didactical Situations in Mathematics’’, by Guy Brousseau (1986). From this theoretical perspective, the referred-to sequence comprises a range of didactical situations which consist themselves in a pattern of interaction with the didactical environment: the activities proposed by the authors of the textbook. And in this sense, the behavior and the strategies adopted by the student as he acts to a situation, reveal the function of the teaching mechanism: the sequence. Or rather, the feedback from the student through the solutions presented as he is confronted with evidences the rise of didactical effects that make the learning of concepts difficult. In the first years of elementary education, the teaching and learning of the aspects concerning rational numbers begin in the building up of the notion of fractions with an approach predominantly meant for whole/part meaning. However, prior researches points that the fraction concept is one of the most difficult to be assimilated by the student, because the effective understanding presupposes articulation of several meanings associated with the rational numbers. The analysis presented in this study comes from checking information obtained in the methodological instruments that were utilized. One of the conclusions of this analysis indicates that the game of didactical variables and the respective values, manipulated by the authors of the textbook in the elaboration of the sequence, influences the strategies mobilized by the students in the resolution of the activities and/or exercises. On the other hand, the investigated didactical sequence seems not to favor the progression of the learning of the students, because it restricts the notion of fractions just to the division of the continuous or discrete whole numbers into “equal” parts. We detected, therefore, that there is a need for an adequacy of the didactical sequence, aiming at the reformulation of the proposed patterns, the revision of the illustrations and texts contained in some activities, the equilibration between the quantity of activities that utilizes the continuous whole numbers and the discrete ones as reference record. And still it is important to point out that the suggested modifications contemplate other connections, such as: the whole/part relation; as well as, alterations that favor the approach and the articulations of other meanings of fractions (quotient, ratio or probability).