Mathematical reasoning fostered by (fostering) transformations of rational number representations

Abstract

In this article we aim to understand the transformations of rational number representations carried out by students and their mathematical reasoning processes. We report part of a Design Based Research, within which an intervention was carried out in a class with 25 students and their teacher, in grades 3 and 4. We analyze six classroom episodes in three main moments concerning the construction of understanding of rational numbers. The results indicate that both transformations and mathematical reasoning processes have an intricate and bidirectional relation, one fostering the other. Students carried out treatments and conversions, including conversions between representations as well as conversions by compositions of different representations. Regarding mathematical reasoning processes, students formulated solving strategies, conjectures and justifications. We also conclude that the social interactions within the class were crucial for the students both doing the transformations and engaging in mathematical reasoning processes.