Exploratory mathematics teaching and the development of students’ use of representations and reasoning processes: An illustration with rational numbers

Abstract

This chapter presents a perspective about the exploratory approach as a possible way to enact inquiry based mathematics teaching. In this approach, the teacher, instead of beginning the class by presenting explanations and examples to the students, proposes them to work on tasks that may lead to the construction of new knowledge. We use as illustration the work of a grade 6 class of students solving tasks involving rational numbers. Our aim is to know how students use representations and reasoning processes, seeking to find out how they deal with different representations and how they formulate generalizations and justifications. We follow a qualitative and interpretative approach, with participant observation of a teaching experiment that included five lessons that were integrally videotaped and transcribed. We analyse episodes from the work of the students in two tasks, one involving a complex relationship between fractions and the other involving the use of fractions as operators. The results show that when solving a task that involves rational numbers given as fractions, the students mostly use the decimal representation, with which they feel rather comfortable. In another task, involving rational numbers as operators, most students use fractions, but some of them also use of decimal numbers and pictorial representations. In both cases, the students chose the representation that they considered best suit their needs. In their written work, the students justify their choices by presenting the computations done when solving a task, adding explanations in natural language. Just by themselves, they are able to use counterexamples to refute a statement, and, during whole class discussions, prompted by the teacher, they are able to make generalizations and justifications based on definitions.