Time(s) in the Didactics of Mathematics: A Methodological Challenge

When the focus of a research study is on the processes during classroom activity, time plays an essential function. It is trivial to observe that -every experiment has to match the time of the school (periods, weekly schedule, holidays); -the processes - either individual or social - develop over time; -observation is carried out over time. The above are three instances of ‘physical time’, the linear sequence of moments measured by the clock . This physical time, which the observer can read on his clock, is in recent videopapers emphasised by the timeline where different media such as video, text, graphics and audio files are synchronised. Yet, Varela himself calls our attention on the existence of an ‘inner time’, that gives rise to human temporality, centred on the present and manifesting as a threefold unity of the just-past and the about-to-occur. This inner time is mostly individual and unconscious. However its features may be inferred from external traces (linguistic expressions, gestures, metaphors). Moreover, it may be partially shaped from outside (e. g. by the teacher), so that the learner becomes conscious of the possibility of moulding it in problem solving. The main purpose of this paper is to show that:1)both kinds of time are relevant in the research in Mathematics Education, when the focus is on the processes of teaching and learning mathematics;2)a further finer specification of both is needed, that requires the introduction of several theoretical constructs related to human temporality and that introduces a lot of methodological problems concerning the relationships between them.