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A networking method to compare theories: metacognition in problem solving reformulated within the Anthropological Theory of the Didactic

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Abstract

An important role of theory in research is to provide new ways of conceptualizing practical questions, essentially by transforming them into scientific problems that can be more easily delimited, typified and approached. In mathematics education, theoretical developments around ‘metacognition’ initially appeared in the research domain of Problem Solving closely related to the practical question of how to learn (and teach) to solve non-routine problems. This paper presents a networking method to approach a notion as ‘metacognition’ within a different theoretical perspective, as the one provided by the Anthropological Theory of the Didactic. Instead of trying to directly ‘translate’ this notion from one perspective to another, the strategy used consists in going back to the practical question that is at the origin of ‘metacognition’ and show how the new perspective relates this initial question to a very different kind of phenomena. The analysis is supported by an empirical study focused on a teaching proposal in grade 10 concerning the problem of comparing mobile phone tariffs.

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Notes

  1. In mathematics, a well-known example of distance between a system and the model used to solve problems appearing in this system is Galois’ theory (using groups and fields) and the problem of solving polynomial equations by radicals. In classical mechanics, the modelling of the solar system, where planets appear as points provided with mass, represents another paradigmatic example of a model that is very distant from the modelled system. It has, however, been extremely efficient to formulate scientific problems related to the solar system and to increase our knowledge about it.

  2. See also Silver & Herbst (2007, p. 62).

  3. Schoenfeld (2007) provides good overview of research results about Problem Solving in the USA during the period 1970 to present.

  4. This is an idea developed by Brousseau in 1986. See Brousseau (1997, pp. 37–40).

  5. This is the formulation used by the OECD in the PISA programme: ‘The assessment is forward-looking, focusing on young people’s ability to use their knowledge and skills to meet real-like challenges, rather than just examining the extent to which they have mastered a specific school curriculum.’

  6. With this assumption, the ATD joins some other visions of human knowledge as the one proposed by the British anthropologist Mary Douglas in her book ‘How institutions think’ (Douglas, 1987).

  7. For more details, see Barbé et al. (2005).

  8. The idea of didactic moment is defined not in a chronological or linear sense, but in the sense of different dimensions (o factors) of the mathematical activity.

  9. For more details on didactic praxeologies and their structuring in didactic moments, see Barbé et al. (2005).

  10. An example of such institutional restrictions in the case of the teaching of limits of functions can be found in Barbé et al. (2005).

  11. From this point of view, the new formulation of Pólya’s problem takes into account the minimal unity of analysis of didactic processes postulated by the ATD: “To describe and interpret didactic phenomena, they have to be related to a sequence of the didactic process including, at least, the process of constructing a local mathematical praxeology” (Bosch & Gascón 2005).

  12. More details about this experimentation are given in Rodríguez (2005).

  13. This proposal is coherent with the evolution of the ‘didactic contract’ as described by Brousseau: “The didactic contract involves the project of its own dissolution. It is understood from the beginning of the didactical relationship that a moment must arrive when it will be broken. At that moment, at the end of the teaching, the taught system will, with the help of the learned knowledge, be assumed to be capable of facing systems without didactical intentions.” (Brousseau, 1997, p. 57).

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Authors and Affiliations

  1. Universidad Complutense de Madrid, Madrid, Spain

    Esther Rodríguez

  2. Universitat Ramon Llull, Barcelona, Spain

    Marianna Bosch

  3. Universitat Autònoma de Barcelona, Barcelona, Spain

    Josep Gascón

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  1. Esther Rodríguez
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  2. Marianna Bosch
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  3. Josep Gascón
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Correspondence to Marianna Bosch.

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Rodríguez, E., Bosch, M. & Gascón, J. A networking method to compare theories: metacognition in problem solving reformulated within the Anthropological Theory of the Didactic. ZDM Mathematics Education 40, 287–301 (2008). https://doi.org/10.1007/s11858-008-0094-z

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