Interpretações dos números racionais: uma análise no 7º ano do ensino fundamental

Abstract

This research follows a qualitative approach, aiming to investigate understandings about interpretations of rational numbers in fractional representation, when activities that emphasize figural records are proposed. From this point of view, the semiotic representation registers proposed by Raymond Duval and the theory of proportional reasoning, developed by Susan Lamon, are adopted as theoretical framework. The text follows a multipaper structure, composed of four manuscripts, with their respective specific objectives (i) analyzing mobilizations of figural registers, linked to rational numbers in fractional representation, with support of the manipulative material Frac-Soma; (ii) investigating understandings about the measure interpretation, through the compensatory principle and the recursive partition principle; iii) exploring concepts related to comparison, ordering and equivalence of rational numbers in fractional representation in approaches of continuous and discrete quantities, when associated with the part-whole interpretation; iv) analyzing understandings about sharing and comparison of quantities through the unitization process and its relations with the quotient and operator interpretations. To meet these objectives, the sources for triangulation of results considered: students' protocols, systematized during the meetings in auxiliary sheets; audio and video recordings that reveal dialogues and gestures that occurred during the process of solving the activities; photographs that reveal moments of manipulation of the Frac-Soma pieces; teacher's/researcher's with reflections on the development of the sequence. Among the results, it is evident that the Frac-Soma, as a manipulative material, contributed to unleash figural records that are associated with operational apprehension, showing mereological and positional changes. Also, the process of successive partitioning of the unit, used in the making of the Frac-Soma, enhanced the acquisition of concepts related to the main notion of rational number in fractional representation, combining evidence of the interpretations part-whole, quotient and measure. Moreover, in the activities related to the measure interpretation, we identified signs of the compensatory principle and the principle of recursive partitioning when we established relations based on the fact that the smaller the unit of measure, the greater the number of units needed, and that whole divisions should consider subunits in accordance with the measure requested. Regarding the part-whole interpretation, the understanding of equivalence relations through the unitization process stands out. On the other hand, in this same interpretation there are difficulties regarding the conservation of area in figures that are not subdivided into parts of the same size, as well as in the process of determining fractions from discrete quantities. Regarding the notions related to sharing, the concepts were understood in a satisfactory manner, involving the necessary partitioning to understand the quotient interpretation. Finally, it should be noted that generalizations were identified from the multiplicative concepts associated with the operator interpretation.