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Semiotic and theoretic control in argumentation and proof activities

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Abstract

We present a model to analyze the students’ activities of argumentation and proof in the graphical context of Elementary Calculus. The theoretical background is provided by the integration of Toulmin’s structural description of arguments, Peirce’s notions of sign, diagrammatic reasoning and abduction, and Habermas’ model for rational behavior. Based on empirical qualitative analysis we identify three different kinds of semiotic actions featuring the organization of the argumentations, and related to different epistemological status of the arguments. In such semiotic actions, the students’ argumentation and proof activities are managed and guided according to two intertwined modalities of control, which we call semiotic and theoretic control. The former refers to decisions concerning the selection and implementation of semiotic resources; the latter refers to decisions concerning the selection and implementation of a more or less explicit theory or parts of it. The structure of the model allows us to pinpoint a dialectical dynamics between the two.

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Notes

  1. According to Peirce (1931–1958), a sign is a triad composed by the sign or representamen (that which represents), the object (that which is represented), and the interpretant: “It [The sign] addresses somebody, that is, creates in the mind of that person an equivalent sign or perhaps a more developed sign. That sign which it creates I call the interpretant of the first sign. The sign stands for something, its object. It stands for that object, not in all respects, but in reference to a sort of idea.” (C.P. 2.228)

  2. Peirce's work is usually referred to in the form C.P. n.m., which means Collected Papers; n, number of volume; m, number of paragraph.

  3. Abduction has already been widely used and discussed in mathematics education, for instance by Arzarello, Micheletti, Olivero, Paola & Robutti (1998), Hoffmann (2005), Radford (2008), and Rivera (2008).

  4. Observe that an abduction is an argument in the sense of Toulmin (see Pedemonte, 2007): the Case is the Claim (possibly with a Qualifier), the Result is the Data and the Rule is the Warrant.

  5. In the classroom, Elementary Calculus is a theory under construction.

  6. It is a case of parallel argumentation, i.e., different arguments are used to support the same conclusion, as described in p. 434 in Knipping (2008).

  7. Recalling that the task we are analyzing is the final one of a teaching sequence, the genesis of this kind of table as a backing for the argumentations can be traced in the previous classroom discussion. In fact, with the guidance of the teacher a table similar to Figure 6 had been collectively built at the blackboard. It worked as a synthesis of the conclusions reached in the various activities on the relationships between the functions graphs and their derivatives and primitives. The mediation of the teacher has certainly played an important role. Though it could be very interesting per se, for the purposes of the paper we do not provide an analysis of the activity from a didactic point of view. We give only some informative elements to help the reader to situate our protocol analysis.

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Acknowledgment

This study is jointly funded by the MIUR and the Università di Torino (PRIN 2007B2M4EK).

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  1. University of Torino, Torino, Italy

    Ferdinando Arzarello & Cristina Sabena

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  1. Ferdinando Arzarello
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  2. Cristina Sabena
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Correspondence to Cristina Sabena.

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Arzarello, F., Sabena, C. Semiotic and theoretic control in argumentation and proof activities. Educ Stud Math 77, 189–206 (2011). https://doi.org/10.1007/s10649-010-9280-3

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