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 selfsustained
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.