MAKING VISIBLE WHAT SEEMS TO BE INVISIBLE IN MATHEMATICAL PROBLEM-SOLVING: Integrating qualitative methods

The main objective of this chapter is to show how the combination of a priori analysis of mathematical tasks, thinking aloud, and protocol analysis methods can generate empirical insights and develop an appropriate unit of analysis for contributing to the theory development of mathematical learning. First, we describe these classical methods and their functional characteristics. Second, according to the socio-cognitive approach these methods are presented as a unit of analysis for studying students’ solving mathematical tasks. Three empirical examples of research with students engaged in a task borrowed from Pisa 2012 items are presented as an application of this methodological approach. Combining the fine-grained analysis of the students’ activity on the task with the examination of their verbal reports allows tracing the dialogic processes associated with the construction of the task. In particular, the task is co-constructed through both an inner dialogue of the student with himself and a social practice embedded in school socialisation. Integrating the three methods could represent a methodological innovation that allows us to re-conceptualise the socio-constructivist approach to mathematical learning and conceive an extended unit of analysis, making visible the role of social practices embedded in everyday school life for mathematical thinking.