How might a 4-year-old pupil come to conceive a line, created by himself with a continuous gesture, as constituted by a series of points? In this paper, we theoretically introduce and discuss a design study of the educational path in which this possibility emerged. The path aimed at leading pupils to experience, in the sense of the enactive perspective and of the sensuous cognition, straight, jagged and curved lines, which arise from their own movements, through particular walking patterns, or via painting and drawing activities.