Linking problem solving and learning contents: The challenge of self-sustained study and research processes

A main difference between the mathematical activity of students and that of researchers is that researchers pursue their mathematical work in a seemingly self-sustaining dynamics of questions and answers, while students rely on teachers to sustain this dynamics. Unlike researchers, students generally do not construct the questions they work on, and do not search, rearrange and question the established contents they need to answer the questions. The basic problem approached in this paper is : could students also engage in a more self-sustaining and complete work with questions and answers ? We first present an analysis of four main paradigms of teaching and learning mathematics, based on different approaches to learners' work with questions and answers. We then discuss and exemplify certain principles for self-sustained mathematical activities using Chevallard's Herbartian schema. The access to new external answers and their test against an appropriate experimental milieu is shown to be a crucial bootstrap for the dynamics of research and study processes.