Abstract
The semiotic approach to mathematics education introduces the notion of “semiotic system” as a tool to describe mathematical activity. The semiotic system is formed by the set of signs, the production rules of signs and the underlying meaning structures. In this paper, we present the notions of system of practices and configuration of objects and processes that complement the notion of semiotic system and help to understand the complex nature of mathematical objects. We also show in what sense these notions facilitate the description and comprehension of building and communicating mathematical knowledge, by applying them to analyze semiotic systems involved in the teaching and learning of some elementary arithmetic concepts.
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Notes
Didactic incident taken from Arrieche’s doctoral thesis (2002).
Rotman (1988) reached a similar conclusion in his semiotic analysis of mathematical activity, when he considered the numbers studied by the Babylonians, Greeks, Romans, and present-day mathematicians are different.
In this case, the “set theory” context refers to the constructions of N based on set coordinability, whereas “axiomatic” refers to Peano’s axiomatics (or other equivalent ones).
In the OSA, we use a weak notion of system as an organized or structured set of elements, which is common in cognitive and social sciences. This allows speaking of “mathematical object” as an entity emerging from the subjects’ system of practices to solve a class of problem situation, mediated by linguistic and material artifacts.
Mathematical objects (both at personal or institutional levels) are, in general, non-perceptible. However, they are used in public practices through their associated ostensive objects (notations, symbols, graphs, etc.). The distinction between ostensive and non-ostensive is relative to the language game (Wittgenstein, 1953) in which they take part.
Institutional objects emerge from systems of practices shared within an institution, while personal objects emerge from specific practices from a person. “Personal cognition” is the result of individual thinking and activity when solving a given class of problems, while “institutional cognition” is the result of dialogue, agreement, and regulation within the group of subjects belonging to a community of practices.
Coherently with the OSA semiotic perspective, we present a triadic characterization of the data, which are analyzed and systematized by a three column array.
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This research work has been carried out as part of the projects SEJ2007-60110/EDUC, MEC-FEDER, and EDU 2009-08120/EDUC.
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Godino, J.D., Font, V., Wilhelmi, M.R. et al. Why is the learning of elementary arithmetic concepts difficult? Semiotic tools for understanding the nature of mathematical objects. Educ Stud Math 77, 247–265 (2011). https://doi.org/10.1007/s10649-010-9278-x
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DOI: https://doi.org/10.1007/s10649-010-9278-x