Reasoning reasonably in mathematics

Abstract

Two tasks designed to encourage mathematical reasoning without any need for calculations were presented to students with the aim of seeking evidence of different forms of attention (Mason, 2003), and using these to learn about the tasks and about students’ power to reason ‘reasonably’ in mathematics. The first task that involves locating a secret place using an applet was solved by two pairs of grade 4 Portuguese students. The second task, involving the structure of magic squares, was proposed to two classes of Portuguese students, aged 12-13. The interactions between pairs of students and teacher probes were taped and transcribed, and, in the second one, students’ written responses were collected as well. In both cases, the data were analysed using the fivefold framework of microstructure of attention. The first case gives evidence that young students can reason ‘reasonably’ but that there are delicate shifts which may require sensitivity on the part of the teacher to help students progress. In the second case, the data analysis shows that these students displayed the power to reason ‘reasonably’ in mathematics, and that difficulties can be accounted for in terms of not only what was being attended to, but the form of that attention.