The authors deconstruct the idea of creativity in its mysterious and romantic dimension on the crossing of three paradoxes; they show, based on teaching situations analyses, that it concerns giving less interest to the creation itself than to conditions of its emergence. The defended idea is: what pupils have to learn is exactly what the teacher cannot teach them and so corresponds to what they have to create.
Learning mathematics is learning a special way of using it when solving new problems. Thus research of conditions of this re-creation of mathematics is in the centre of teacher's activity.