Abstract
This paper illustrates two examples of contexts for approaching validation, framed by a research program on semiotic mediation in the mathematics classroom. The question of how to approach theoretical reasoning, a major problem in mathematics education, is especially difficult when physical, truly tangible, artefacts are in play. Whenever a student is given a mathematical task to solve by using some artefact, in the solution process the student displays intense, observable semiotic activity (gazing, gesturing, writing, speaking, drawing and so on). A major aim of mathematics teaching is to foster the students’ construction of the relationship between those produced signs and mathematical signs. The two examples show the successful introduction of physical artefacts at both the primary and secondary levels as tools of semiotic mediation in the process of mathematical validation.
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Notes
- 1.
The availability of mathematical machines in a mathematics classroom cannot be taken for granted. Hence, there is the risk that such a teaching experiment cannot be reproduced for lack of tools. This is the main reason why some years ago the MMLab was opened to classrooms, under the guidance of laboratory operators. The person responsible for this activity is Michela Maschietto. The activity has been designed in order to offer a 2-h reconstruction (a short one) of the classroom experiments with mathematical machines. An average of 1300–1500 secondary students a year come with their mathematics teacher to experience the mathematics laboratory hands-on. These numbers are demanding, yet represent a tiny proportion of the whole population. Hence, our research group aims at disseminating this activity by offering schools travelling exhibitions, ready-made kits and work-sheets. A long documentary on a typical classroom visit (in Italian), broadcast by the national network RAIeducational (Explora scuola), is available at http://www.explora.rai.it/online/amministrazione/uploads/asx/97302_exp.asx.
- 2.
In a short visit (less than 2 h), the exploration of three-dimensional models is carried out by the laboratory operator during the historical introduction.
- 3.
The interested reader may download a Cabri simulation from the website (in Italian): http://associazioni.monet.modena.it/macmatem/lauree%20sc/Caval.htm, by clicking on “simulazione” on the right.
- 4.
In a mathematics classroom, if more time is allotted, more freedom can be left for students’ exploration.
- 5.
In Fig. 13, the point A and the length of the bar KE are fixed; K is dragged back and forth in the rail HL, pulling KEB and forcing the fissured side BA of KBA to rotate around A. Fig. 13a is taken from the exploration sheet. Fig. 13b shows the tool in another state, after a short sliding of K on the horizontal rail HL with the dependent rotation of BA around A; also the path of B during the motion (i.e. an arc of parabola) has been drawn, i.e. the same drawing that students produce in the step 6. Fig. 13b is not taken from the exploration sheet, but is added here for the reader’s understanding: the same letters as in Fig. 12 have been used for the sake of clarity.
- 6.
This exploration sheet has been designed and tested by the staff of the Laboratory of Mathematical Machines (Michela Maschietto with Carla Zanoli, Rossana Falcade, Francesca Martignone).
- 7.
It is beyond the scope of this chapter to analyze the similarities and differences that emerge in the exploration of ancient tools and present dynamic geometry environments.
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Acknowledgments
Research funded by MIUR (PRIN 2007B2M4EK: “Instruments and representations in the teaching and learning of mathematics: theory and practice”).
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Bussi, M.G.B. (2010). Historical Artefacts, Semiotic Mediation and Teaching Proof. In: Hanna, G., Jahnke, H., Pulte, H. (eds) Explanation and Proof in Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0576-5_11
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