Aspectos do raciocínio proporcional de alunos do 7º ano mobilizados em experiências dialógicas de resolução e de elaboração de tarefas matemáticas

Abstract

This study aimed to answer the following research question: How does the solving and designing of tasks on proportionality, by 7th-grade students in Middle School, through dialogic experiences in lessons based on the Exploratory Teaching approach, enable the mobilization of aspects of proportional reasoning? The research adopts a qualitative approach, focusing on understanding students' thinking processes during the design and solving of mathematical tasks. The data analyzed consist of dialogues between the researcher and small groups of students, as well as the written productions and records created by them. The dissertation is organized in a multipaper format, composed of an extended introduction and three articles, which, in an articulated manner, provided partial answers that contributed to addressing the main research question. The first article aims to understand how the solving of an exploratory task, through dialogic experiences with eight groups of 7th-grade Middle School students, can contribute to the mobilization of aspects of proportional reasoning. The results showed that, when solving an exploratory task on proportional relationships, students were able to mobilize important aspects of proportional reasoning, such as quantity comparison, multiplicative comparisons, and, to a lesser extent, covariation. The second article seeks to understand how the designing of mathematical tasks by 7th-grade Middle School students, based on a dialogic perspective, can contribute to the structuring process of the proportional relationship between two quantities and to the recognition of the multiplicative nature present in this relationship. The analysis of recorded dialogues, the tasks designed, and the students' own solutions to their tasks indicated that, despite initial difficulties, the groups were able, at different levels, to mobilize an understanding of the multiplicative nature between quantities. The third article aimed to understand, from a dialogic perspective, the mobilization of progressive and regressive reasoning in the design of tasks involving proportional relationships by 7th-grade Middle School students. The analyses, as in the second article, revealed difficulties in structuring and understanding these relationships. However, after dialogic interventions carried out by the researcher, the groups were able to design their tasks and, in their solutions, demonstrated the mobilization of both progressive and regressive reasoning. The articulation of the results from the three articles allowed the conclusion that, compared to task solving, task design proved to be a more challenging process for both students and the teacher. Nevertheless, it showed great potential for mobilizing and developing aspects of proportional reasoning. Additionally, it fostered the development of dialogic interactions, which were essential for the researcher to identify students’ zones of proximal development and to carry out specific interventions and questions that stimulated students’ thinking and reflection. Finally, this research can serve as a springboard for further studies involving the design of mathematical tasks by students, whether aimed at fostering the understanding of other mathematical concepts or at mobilizing other aspects of proportional reasoning.