Social activity method: a fractal language for mathematics. (English)

Philos. Math. Educ. J. 29, 23 p., electronic only (2015).

Summary: In this paper I shall present and develop my organisational language, “social activity method" (SAM) and illustrate some of its applications. I shall introduce a new scheme for “modes of recontextualisation" that enables the analysis of the ways in which one activity ‒ which might be school mathematics or social research or any empirically observed regularity of practice ‒ recontextualises the practice of another and I shall also present, deploy and develop an existing scheme ‒ “domains of action" ‒ in an analysis of school mathematics examination papers and in the structuring of what I refer to as the “esoteric domain". This domain is here conceived as a hybrid domain of, firstly, linguistic and extralinguistic resources that are unambiguously mathematical in terms of both expression and content and, secondly, pedagogic theory ‒ often tacit ‒ that enables the mathematical gaze onto other practices and so recontextualises them. A second and more general theme that runs through the paper is the claim that there is nothing that is beyond semiosis, that there is nothing to which we have direct access, unmediated by interpretation. This state of affairs has implications for mathematics education. Specifically, insofar as an individual’s mathematical semiotic system is under continuous development ‒ the curriculum never being graspable all at once ‒ understanding ‒ as a stable semiotic moment ‒ of any aspect or object of mathematics is always localised to the individual and is at best transient and the sequencing of such moments may well also be more individualised than consistent in some correspondence with the sequencing of the curriculum. This being the case, a concentration on understanding as a goal may well serve to inhibit the pragmatic acquisition and deployment of mathematical technologies, which should be the principal aim of mathematics teaching and learning. The paper is primarily concerned with mathematics education. SAM, however, is a language that is available for recruiting and deploying in potentially any context as I have attempted to illustrate with some of the secondary illustrations in the text.