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The aim of this paper is to investigate the progressive manner in which students gain fluency with cultural algebraic modes of reflection and action in pattern generalizing tasks. The first section contains a short discussion of some epistemological aspects of generalization. Drawing on this section, a definition of algebraic generalization of patterns is suggested. This definition is used in the subsequent sections to distinguish between algebraic and arithmetic generalizations and some elementary naïve forms of induction to which students often resort to solve pattern problems. The rest of the paper discusses the implementation of a teaching sequence in a Grade 7 class and focuses on the social, sign-mediated processes of objectification through which the students reached stable forms of algebraic reflection. The semiotic analysis puts into evidence two central processes of objectificationiconicity and contraction.

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The goal of this article is to present a sketch of what, following the German social theorist Arnold Gehlen, may be termed "sensuous cognition." The starting point of this alternative approach to classical mental-oriented views of cognition is a multimodal "material" conception of thinking. The very texture of thinking, it is suggested, cannot be reduced to that of impalpable ideas; it is instead made up of speech, gestures, and our actual actions with cultural artifacts (signs, objects, etc.). As illustrated through an example from a Grade 10 mathematics lesson, thinking does not occur solely in the head but also in and through a sophisticated semiotic coordination of speech, body, gestures, symbols and tools.

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This paper is a commentary on the problem of networking theories. My commentary draws on the papers contained in this ZDM issue and is divided into three parts. In the first part, following semiotician Yuri Lotman, I suggest that a network of theories can be conceived of as a semiosphere, i.e., a space of encounter of various languages and intellectual traditions. I argue that such a networking space revolves around two different and complementary ''themes''-integration and differentiation. In the second part, I advocate conceptualizing theories in mathematics education as triplets formed by a system of theoretical principles, a methodology, and templates of research questions, and attempt to show that this tripartite view of theories provides us with a morphology of theories for investigating differences and potential connections. In the third part of the article, I discuss some examples of networking theories. The investigation of limits of connectivity leads me to talk about the boundary of a theory, which I suggest defining as the ''limit'' of what a theory can legitimately predicate about its objects of discourse; beyond such an edge, the theory conflicts with its own principles. I conclude with some implications of networking theories for the advancement of mathematics education.

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In this article, we present a sociocultural alternative to contemporary constructivist conceptions of classroom interaction. Drawing on the work of Vygotsky and Leont'ev, we introduce an approach that offers a new perspective through which to understand the specifically human forms of knowing that emerge when people engage in joint activity. To this end, we present two concepts: space of joint action and togethering. The space of joint action allows us to capture the collective and sensuous or intercorporeal dimensions of thought and feeling in interaction. We resort to the concept of togethering to capture the ethical commitment participants make to engage in and produce activity. These concepts are illustrated through a discussion of concrete episodes from an elementary mathematics classroom.