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The Planimeter as a Real and Virtual Instrument that Mediates an Infinitesimal Approach to Area

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Digital Technologies in Designing Mathematics Education Tasks

Part of the book series: Mathematics Education in the Digital Era ((MEDE,volume 8))

Abstract

Drawing on a didactic gap detected between the elementary concept of area and the infinitesimal approach to it within the Italian secondary school curriculum, the notion of swept area is introduced in grades 10–11. The idea of swept area is introduced through the mediation of an artifact, the Polar Planimeter, both as a concrete physical-tool and as a virtual-object. It triggers and supports the semiotic productions of the students so that they can grasp the new concept. The notion of didactic cycle is used for designing students’ learning sequences. The activities in such sequences are of two types: sensory-motor and symbolic. The mediation of the artifact allows intertwining the two types so that the one can constantly be built on the other. Indeed, the practices mentioned above show a deep intertwining between their cultural and cognitive components.

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Notes

  1. 1.

    The Italian expression is similar but not identical with English expressions like “the more … the better”; we leave it since, contrary to what can happen in English, it emphasizes the perfect balance between the two sides of the comparisons with the words and their sounds. This effect can be lost or smoothed over in English.

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Authors and Affiliations

  1. Dipartimento di Matematica “G. Peano”, Università di Torino, Turin, Italy

    Ferdinando Arzarello

  2. Istituto Sociale and Scuola Secondaria di I grado E. Artom, Turin, Italy

    Daniele Manzone

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  1. Ferdinando Arzarello
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  2. Daniele Manzone
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Correspondence to Daniele Manzone .

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Editors and Affiliations

  1. Department of Education Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China

    Allen Leung

  2. Department of Mathematics “G. Castelnuovo”, “Sapienza” University of Rome, Rome, Italy

    Anna Baccaglini-Frank

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Arzarello, F., Manzone, D. (2017). The Planimeter as a Real and Virtual Instrument that Mediates an Infinitesimal Approach to Area. In: Leung, A., Baccaglini-Frank, A. (eds) Digital Technologies in Designing Mathematics Education Tasks. Mathematics Education in the Digital Era, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-43423-0_7

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  • DOI: https://doi.org/10.1007/978-3-319-43423-0_7

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  • Print ISBN: 978-3-319-43421-6

  • Online ISBN: 978-3-319-43423-0

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