Rational number interpretations as one of the central elements in the development of proportional reasoning: an approach with Frac-Soma

Abstract

This article aims to analyze understandings of students of the 7th grade of elementary school when solving activities that emphasize the rational number interpretations, quotient and operator. For this purpose, a qualitative approach was employed to assess 11 activities that utilized Frac-Soma and were conducted with a class of ten students from a public school in Sobradinho, RS. Data production considered protocols, recordings (audio and video), photographs, and the teacher/” ’researcher’s logbook. Among the results was that notions related to fair sharing were understood, as the students established connections between the requested quantities and the partitioning process, necessary for understanding the quotient interpretation and the development of proportional reasoning. On the other hand, obstacles were identified in relation to the comparison notion, as conclusions were not systematized regarding the notions of “the more”/“the less” one quantity is greater/smaller than another. To minimize these difficulties, during the teacher/” ’researcher’s interventions, it was necessary to emphasize the unitization process, which is fundamental to understanding the equivalence notion and this to proportional reasoning. In activities involving operator interpretation, it was found that in the action of partitioning the integer, as well as “exchanging” Frac-Soma parts for others, there was a “loss” of the unit reference. After discussions in the groups and teacher/researcher interventions, signs of understanding that the operator is a function capable of transforming the unit into another similar one was noticed. Thus, it could be concluded that the students presented an understanding of these rational number interpretations, although the notions of operator and comparison are still a challenge for some.