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Connectedness of Problems and Impasse Resolution in the Solving Process in Geometry: A Major Educational Challenge

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International Perspectives on the Teaching and Learning of Geometry in Secondary Schools

Part of the book series: ICME-13 Monographs ((ICME13Mo))

Abstract

Our contribution shows the anticipated effect of what we call connected problems in developing the competencies of students and their acquisition of mathematical knowledge. Whilst our theoretical approach focuses on didactic and cognitive interactions , we give special attention to a model to reason about learners’ conceptions, and the ideas of mathematical working space and zone of proximal development, in order to explore how connected problems can help to resolve moments of impasse of a student when solving a proof problem in geometry . In particular, we discuss how the notion of interaction moves our theoretical framework closer to the methodological challenges raised in the QED-Tutrix research project jointly being realized in didactics of mathematics and computer engineering.

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Notes

  1. 1.

    For a comparative study and a complete update on related tutorial systems, see Tessier-Baillargeon, Leduc, Richard, and Gagnon (2017).

  2. 2.

    For example, if we know that a student cannot solve a problem because he or she does not confront the hypotheses, then we ask him or her a new problem in which the main issue is the discovery of incompatibility between hypotheses.

  3. 3.

    Deductive isles is our translation from the French îlot déductif that considers the network of mathematical properties and definitions accepted or actually used in a given class, which includes the implicit hypothesis and the inferential shortcuts tolerated in the didactic contract.

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Acknowledgements

This research has been made possible by a grant from the Conseil de Recherches en Sciences Humaines (CRSH 435-2015-0763), Gouvernement du Canada.

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Authors and Affiliations

  1. Université de Montréal, Montréal, Canada

    Philippe R. Richard

  2. École Polytechnique de Montréal, Montréal, Canada

    Michel Gagnon

  3. Universitat Autònoma de Barcelona, Barcelona, Spain

    Josep Maria Fortuny

Authors
  1. Philippe R. Richard
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  2. Michel Gagnon
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  3. Josep Maria Fortuny
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Corresponding author

Correspondence to Philippe R. Richard .

Editor information

Editors and Affiliations

  1. School of Education, University of Michigan, Ann Arbor, Michigan, USA

    Patricio Herbst

  2. Institute of Teacher Education, Penang, Malaysia

    Ui Hock Cheah

  3. Département de didactique, Université de Montréal, Montreal, Québec, Canada

    Philippe R. Richard

  4. School of Education, University of Southampton, Southampton, United Kingdom

    Keith Jones

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Richard, P.R., Gagnon, M., Fortuny, J.M. (2018). Connectedness of Problems and Impasse Resolution in the Solving Process in Geometry: A Major Educational Challenge. In: Herbst, P., Cheah, U., Richard, P., Jones, K. (eds) International Perspectives on the Teaching and Learning of Geometry in Secondary Schools. ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-77476-3_20

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  • DOI: https://doi.org/10.1007/978-3-319-77476-3_20

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