Graus de não congruência semântica nas conversões entre os registros geométrico bidimensional e simbólico fracionário dos números racionais

Abstract

This research proposes to categorize the degrees of non-congruence semantics in the conversion between the registers, two-dimensional geometric and fractional symbolic of rational numbers. For this, we take as a reference the research of Duval (2004) to propose a classification of the types of semiotic representations of the register of two-dimensional geometric representation of rational numbers, based on the visual, dimensional and qualitative variables of the geometric figures. Also, we have as a theoretical mark the Durval’s studies (1994, 2004, 2012b) to identify the types of geometrical appreciations that are required in conversions between these records and the criterion of semantic congruence, defined in Duval (2004, 2009, 2011). Our research is composed of two stages, the first one comprised of an analysis of the characteristics and treatments specific to each register, and the proposition of the previous model of categorization of degrees of non-congruence semantic between these conversions, based on our theoretical reference. The second stage corresponds to the empirical study, for validation of the model, accomplished with a total of 381 students, belonging to the 6th and 9th year of Elementary School and 1st and 3rd years of High School of five schools of the State Education Network of Alagoas, located in the city of Maceió. The empirical research was carried out with the application of a research instrument, containing 12 items in which the conversion of the two-dimensional geometric register to the fractional symbolic was requested, as well as a clinical-critical interview with some individuals participating in the research. Six degrees of non-semantic congruence were categorized in conversions that had as a starting register the two-dimensional geometric and as arrival register the fractional symbolic of rational numbers. We conclude that the degree 1 of non-congruence semantics involves the geometric figures classified in our study as perceptual with an integer, which only require the perceptual and discursive apprehensions of its figurative units and are well adapted to the double counting procedure. The degree 2 of non-congruence semantics involves the perceptual figures with more than one integer and also, as in the previous level, only need the perceptual and discursive apprehensions of their figurative units; however, have some units that do not correspond semantically with the symbolic units, leaving these conversions with a higher level of difficulty than the previous one. The degree 3 of non-congruence semantics includes the figures classified as operative by inclusion of the parts. At this level it is possible to perform a figural treatment to obtain, in the conversion to the fractional symbolic register, an irreducible fraction. In grade 4 of non-congruence semantics, the geometric figures are the operative ones by division. It is since that level that the figurative treatments are indispensable for the conversion to take place, that is, in addition to the perceptual and discursive apprehensions, the operative apprehension is required. In degree 5 of non-congruence semantics, the geometric figures are classified into operative by deconstruction of the parts. Although they have parts or subfigures with congruent areas, their forms are heterogeneous, making it difficult to see the congruence of the areas between the parts. Finally, degree 6 of non-congruence semantics involves operative figures by deconstruction of areas and forms, which require a figural treatment of higher cognitive cost.