Towards a better understanding of hands-on approaches in maths education: a reflection from compared experiences in higher, secondary and primary education

The aim of this work is to show and compare some didactical experiences addressed to primary school children as well as to grown up students, up to higher education, based on a hands-on approach to mathematics. The experiment in primary school was inspired by the work of Mary Everest Boole and involves extremely simple and common tools aimed to create an intimate relationship between the child and numbers and shapes. This gives the opportunity to connect abstract concepts to experience, communicating mathematical ideas and nourishing the unconscious mind to prepare it to science. The point of view of Mary Boole was to introduce the geometry of curves to children, going beyond concepts like lines and circles. By using wires, children deal with superior concepts, like that of tangent lines and curve envelopes and through the movement of the hands and visualization they witness the generation of a curved shape. In the contest of secondary school and higher education, an elective course on construction and use of machines to draw curves has been proposed, the level appropriately adapted either to high school students or to universitary ones. The main goal was to present an informal way to learn mathematics, aimed to connect the analytic representation of curves, to their drawing, and their shape. The process of learning goes through the manipulation, and the physical action of changing the setting of the drawing machine, which reflects on the change of the shape of the plotted curve The tactile and visual experience induces an attitude of curiosity and wishing for knowledge: the machine makes the student an active player, the tracing of the curve is in some sense anytime new, since it involves movement, visualization, reasoning in a dynamical way. Students who were not particularly motivated and passionate in studying mathematics achieved excellent results.