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A semiotic perspective on mathematical activity provides a way of conceptualizing the teaching and learning of mathematics that transcends and encompasses both psychological perspectives focussing exclusively on mental structures and functions, and performance-focussed perspectives concerned only with student's behaviours. Instead it considers the personal appropriation of signs and the underlying meaning structures embodying relationships between signs. It is concerned with patterns of sign use and sign production, including individual creativity in sign use, and the underlying social rules and contexts of sign use. It is based on the concept of a semiotic system, comprising signs, rules of sign production, and an underpinning meaning structure. This theorisation is applied to the learning of number, from counting to calculation. Historical, foundational and developmental (i.e., learning) perspectives are explored and contrasted. It is argued that in each of these domains, the dominant significant activity concerns the production of sequences of signs.

Mathematical signs and symbols have a decisive role for coding, constructing and communicating mathematical knowledge. Nevertheless these mathematical signs do not already contain mathematical meaning and conceptual ideas themselves. The contribution will present basic elements of an epistemology of mathematical knowledge and then apply these theoretical ideas for analyzing case studies of two teaching episodes from elementary mathematics teaching. In this way different roles of mathematical signs as means of communication (oral function), of indicating (deictic function) and of writing (symbolic function) will be elaborated.

This comment attempts to identify different "semiotic perspectives" proposed by the authors of this special issue according to the problems they discuss. These problems can be distinguished as problems concerning the representation of mathematical knowledge, the definition and objectivity of meaning, epistemological questions of learning and activity in mathematics, and the social dimension of sign processes. The contributions are discussed so as to make visible further research perspectives with regard to "semiotics in mathematics education".

The article investigates in the first part critically dyadic and essentialist understanding of signs and utterances in mathematics and mathematics education as opposed to a triadic view. However even Peircean semiotics, giving priority to triadic, dynamic sign may face challenges, such as explaining the sign as a pragmatic act and how signs are related to context. To meet these and other hurdles an explicit communicational, pragmatic and triadic view, found in parts of the works of Buhler, Bakhtin, Habermas, and Halliday, is developed. Two basic principles are combined and established in a theoretical framework. Firstly, whenever uttering, there will exist in any semiotic sign system, dynamic reciprocity and simultaneity between expressing through form, referring to content, and addressing as an act. Secondly, meaning will be created by the dynamics between given and new in utterances and between utterances and contextual genres. The latter principle explains how meaning merge in communication dynamically and create the basis for a discursive understanding of semiosis and hence even learning at large. The second part exemplifies each of the three main aspects and the dynamics of utterance and genre and given and new by excerpts from a textbook in mathematics education. The concept "positioning", in use for operationalisation, is explained in relation to main principles of the framework. The article ends focusing crucial implications for validation when moving from a dyadic to a triadic understanding of mathematics and mathematics education.

Learning is better than knowing, generalization is more illuminating than abstract generality or universality because we perceive and thus become conscious of change or development only. Signs and representations establish the dialectic of fixation on the one hand and transformation on the other, which is so essential to learning and cognition. Mathematical epistemology from a semiotic point of view therefore is above all a genetical epistemology. All real mathematical activity is concerned with representations of mathematical entities rather than with things in themselves and with the processes of continuous transformation of a given representation into others. This paper tries to give an overview of the essential relationships between activity theory, epistemology and mathematical education, using the semiotics of Charles S. Peirce as a unifying reference. It is certainly beyond the scope of such a paper to spell out all the questions involved in every detail. Much of what is said in the four short sections to follow is calling for further concretization and research.

The aims of this article are to document the types of signs that young children make to represent their mathematical thinking, and to determine the extent to which features of usage-based language acquisition are evident in children's early graphical communications made in mathematical contexts. Studies of young children's symbolic principles in ontogeny and research into the acquisition and development of language provide insights into the rich foundational knowledge on which they build their early mathematical inscriptions. The study conceives of children's mathematical abstractions as emergent cognitive representations, originating in their need to communicate within personally meaningful contexts. The collected ethnographic data comprise mathematical inscriptions from seven children aged three to four years in their nursery school and written observations from their teachers and the first researcher. Analysis follows an interpretive, social-semiotic paradigm; the inscriptions were analysed to show how they convey their emerging mathematical understandings, and how this supports their emergent abstractions. The findings illuminate children's strategies as they communicate their thinking, indicating the importance of symbolic number knowledge in acquiring the abstract graphical language of mathematics. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]