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Objects, signs, and representations in the semio-cognitive analysis of the processes involved in teaching and learning mathematics: A Duvalian perspective

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Abstract

In mathematical activities and in the analysis of mathematics teaching-learning processes, objects, signs, and representations are often mentioned, where the meaning assigned to those words is sometimes very broad, sometimes limited, other times intuitive, allusive, or not completely clear. On the other hand, as international research in mathematics education has shown, the confusion between objects, signs, and representations is one of the main reasons of the difficulties in learning mathematics. But what kinds of objects are involved in teaching-learning mathematics? Why should we distinguish a knowledge object, and in particular a mathematical object, from one of its representations? What is meant by “sign”? Can we equate the term “sign” with the term “representation”? In this article we will try to provide an answer to these questions, taking into account the main contributions to mathematics education made by the semiotic theories that are considered the most relevant in the analysis of the cognitive processes involved in mathematical activities. In particular, we will refer to the semiotic representation registers theory, on which Duval’s semio-cognitive approach is based. In general it will be shown that the choice of a semiotic approach to mathematics education assumes a fundamental theoretical choice closely tied to the fundamental distinction between classifying signs and classifying semiotic systems, which is often implicit or rather not emphasized enough. The example presented shows how the semio-cognitive analysis of the processes involved in the solution of a mathematical problem provides new and effective professional reading keys of students’ difficulties in learning mathematics.

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Notes

  1. On the subject of representation and the issues it raises in teaching-learning mathematics, an important collection of points of view, studies, and research predating the 90’s is that of Janvier (1987), most of all within the scope of psychology and pedagogy.

  2. The translation of all quotes from non-English-language sources has been rendered by the author.

  3. A practice is any action carried out by someone to solve a mathematical problem, communicate the solution to other persons, validate and generalize the solution in other contests and problems (Godino & Batanero, 1998).

  4. On the contemporary cultural-historical theories that include the category of activity see: Roth & Radford (2011).

  5. The word “objectification,” from “objectify,” derives from Latin obiectum (“that which is placed in front” or “that which is thrown against”) and facere (“to make”). It therefore means etymologically “to place something in front of someone in order to make it apparent, i.e., present to the senses.”

  6. A = Gottfried Wilhelm Leibniz, Sämtliche Schriften und Briefe, herausgegeben von der Preussischen (Deutschen) Akademie der Wissenschaften zu Berlin, Reihe 6. Darmstadt 1923, Leipzig 1938, Berlin 1950 - (followed by volume, part, page).

  7. In Über Sinn und Bedeutung, when referring to the principle of substitution, Frege cites Leibniz: “Eadem sunt, quae sibi mutuo substitui possunt, salva veritate” (Frege, 1892, p. 35). (For further details, see: Bagni, 2006; D’Amore, Fandiño Pinilla, Iori, & Matteuzzi, 2015).

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  1. NRD (Nucleo di Ricerca in Didattica della Matematica), University of Bologna, Bologna, Italy

    Maura Iori

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  1. Maura Iori
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Iori, M. Objects, signs, and representations in the semio-cognitive analysis of the processes involved in teaching and learning mathematics: A Duvalian perspective. Educ Stud Math 94, 275–291 (2017). https://doi.org/10.1007/s10649-016-9726-3

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