Abstract:

One of the main problem in teaching mathematics is to make people interested and active in the subject which is usually considered difficult and boring.
A possible approach is to propose some laboratories where students can get in touch with maths investigating, constructing and using logic and fantasy.
In each laboratory, some aspects of a mathematical topic (motivations, main ideas, results and historical anecdotes) are proposed from an elementary, practice and amusing point of view. Moreover, students are invited to recognize the proposed topic in art and nature.
The strategy is to use common, coloured, and funny materials (paper, glue, soap and water, plugs, balloons) to make models of the mathematical topic and enjoy with it.
The pedagogical purpose is to make student able to formulate possible solutions for a given problem (always proposed disguised as games and verified practically), to discuss starting conditions (hypotheses) and to justify the claim (ideas of proof). In this way mathematics are not proposed as a theory constructed abstractly by skill people but problems are proposed and students discover a solution by themselves.
We describe, as examples, three laboratories that we proposed in Italy in some primary and high schools. Depending on the age of students one can adapt them to develop different aspects.
1) Soap bubbles (minimal surfaces theory).
Mathematical Aims. Students use the experimental method (prevision, experimental verification, discussion and justifications of the results). They understand the differences between proving a claim in maths and verifying conjectures in sciences. They are introduced in the theory of minimal surfaces.
Interdisciplinary Character . Minimal surfaces in sculpture and architecture.
2) Where and how far are you? (Jordan Theorem and the notion of distance).
Mathematical Aims. Students analyze the behaviour of closed simple curves on different surfaces (plane, sphere, Moebius band). They discuss the Jordan’ s Theorem which holds in some cases but not in all, so to understand the importance of the hypotheses. Via proposed challenges, they are forced to select carefully the surface for a problem to have a solution. In the second part of the laboratory, at the end they understand that there are different ways to compute distances (many rules) in the plane and they draw the same metric locus changing the used distance.
Interdisciplinary Character. Moebius band in art and in technology, geography, topography.
3)Bees and fishes; pin, mirror and knitting needle (Tessellations, polygons, symmetries).
Mathematical aims. In the first part, students discover what regular polygons (and irregular figures) can be used to floor the whole plane. They find methods to compute directly internal angles of regular polygons and to construct “by hands” central, axial and rotational symmetries (using pin, mirror and knitting needle).
Interdisciplinary Character. Tessellation in nature (bees, flowers, pineapple), in art (Alhambra of Granada, the Egg statue in Vegreville ,Canada, the Escher drawings). Tessellations in technical design.

We presented also these labs’ in Vercelli (Politecnico di Torino, II Faculty of Engineering) during the event “Researchers Stars for One Night” (supported by the European Union) in the last three editions, obtaining a big success.

Keywords:

math laboratories, tessellations, soap bubbles and minimal surfaces, jordan theorem.