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Proofs, Semiotics and Artefacts of Information Technologies

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Book cover Explanation and Proof in Mathematics

Abstract

Recently, as part of a larger research project, we carried out a long-term teaching experiment in the ninth and tenth grades of a scientific high school aimed at getting students to approach issues of validation and of teaching of mathematical proof. Assuming a Vygotskian perspective, we focused on the social construction of knowledge and on semiotic mediation as accomplished by the teacher through the use of cultural artefacts. This paper discusses some results from the teaching experiment. It aims to clarify the role of information technologies in introducing students to a theoretical perspective. The first part of the paper introduces the construct of semiotic potential of an artefact. This construct is part of a model that was developed as result of the teaching experiment. Such a model aimed to describe the functioning of semiotic mediation in teaching – learning processes centred on the use of an artefact. The second part discusses the use of two different artefacts of information technology, a dynamic geometry environment and a software package for the teaching of algebra. The results indicate that both artefacts were successful semiotic mediators in the classroom and helped students understand how one arrives at mathematical validation.

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Notes

  1. 1.

    Actually a DGS provides a larger set of tools, including for example “measure of an angle”, “rotation of an angle” and the like, which implies that its whole set of possible constructions is larger than that attainable only with ruler and compass, (see Stylianides and Stylianides 2005, for a full discussion).

  2. 2.

    This definition has been widely used and further elaborated, generating subsequent interpretation models for both conjecturing and proving (Mariotti 2006; Pedemonte 2002; Mariotti and Antonini 2007; Antonini and Mariotti 2008).

  3. 3.

    An exception is mathematical induction, which is explicitly treated, and to which a specific training is devoted. However, very rarely is mathematical induction presented in comparison to other modalities of proving, which are commonly considered natural and spontaneous ways of reasoning.

  4. 4.

    See the current literature. Using the operational-structural terminology of Sfard (1991), one can say that the operational character transforming algebraic formula and expressions shows to be persistent, while the absence of “structural conceptions” appears evident (Kieran 1992, p. 397). On the contrary, a structural conception becomes crucial in order to grasp the meaning of “symbolic calculation”, in particular if one considers the change that the term ‘calculation’ has to achieve when passing from the numerical to the algebraic context.

  5. 5.

    For instance, the statement a+b=b+a. For brevity reasons, I will not enter into details in the description of axioms and definitions of the Theory. I would rather concentrate on the meta-theoretical aspects.

  6. 6.

    Recall that by Theory we mean set of axioms, definitions and theorems that have a counterpart in the collection of Buttons available in the microworld.

  7. 7.

    Because of its reference to specific sets of axioms, any pre-defined Theory cannot be modified.

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Acknowledgments

Research funded by MUIR (PRIN 2005019721: Meanings, conjectures, proofs: from basic research in mathematics education to curricular implications).

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Authors and Affiliations

  1. Dipartimento di Scienze Matematiche ed Informatiche, Università di Siena, Piano dei Mantellini 44, Siena, 53100, Italy

    Maria Alessandra Mariotti

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  1. Maria Alessandra Mariotti
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Correspondence to Maria Alessandra Mariotti .

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Editors and Affiliations

  1. Ontario Institute for Studies in, University of Toronto, Bloor Street W. 252, Toronto, M5S 1V6, Canada

    Gila Hanna

  2. FB Mathematik, Universität Duisburg-Essen, Essen, 45117, Germany

    Hans Niels Jahnke

  3. Inst. Philosophie, Universität Bochum, Universitatsstr. 150, Bochum, 44801, Germany

    Helmut Pulte

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Mariotti, M.A. (2010). Proofs, Semiotics and Artefacts of Information Technologies. 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_12

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