Movements involving mathematical reasoning processes mobilized by students entering Higher Education in an exploratory task

Abstract

This article analyzes movements involving processes of mathematical reasoning mobilized by students entering Higher Education, taking the subject Differential and Integral Calculus (DCI), in discussions based on an exploratory task. The research is qualitative and interpretative, and the data collected for analysis are composed of (i) protocols containing written records of the discussions and (ii) audios of the small group discussions held in a regular Engineering class. The discussions of four groups, composed of 3 students in each group, were analyzed. The results obtained showed, throughout the discussion excerpts of the groups, a cyclical movement, with back and forth, of reasoning about mathematical relations and developing statements. Students raise conjectures and, when conjecturing, three distinct movements were identified, which are: (i) abandoning conjectures definitively or provisionally; (ii) refuting such conjectures with justifications or without justifications; or still (ii) accepting the conjectures seeking to recognize and explain the validity (or not) of these statements by justifying them. In some moments, this movement culminated with extending, to more general situations, the regularities observed in particular cases, which led to the process of generalizing.

Mathematical Teaching; Teaching Differential and Integral Calculus; Processes of mathematical reasoning; Exploratory tasks