Emerging insights from the evolving framework of structural abstraction

ion seems to have gained a bad reputation because of the criticism articulated by the situated cognition (or situated learning) paradigm, and, as a consequence, has almost disappeared. This criticism rests primarily on traditional approaches considering abstraction as decontextualization and often confusing abstraction with generalization. The recent contribution by Fuchs et al. (2003) shows that such classical approaches to abstraction still exist: “To abstract a principle is to identify a generic quality or pattern across instances of the principle. In formulating an abstraction, an individual deletes details across exemplars, which are irrelevant to the abstract category [...]. These abstractions [...] avoid contextual specificity so they can be applied to other instances or across situations.” (p. 294) However, scholars in mathematics education argued against the decontextualization view of abstraction. Van Oers (1998, 2001), for instance, argued that removing context must impoverish a concept rather than enrich it. Several other scholars have reconsidered and advanced our understanding of abstraction in ways that account for the situated nature of knowing and learning in mathematics. Noss and Hoyles (1996) introduced the notion of situated abstraction to describe “how learners construct mathematical ideas by drawing on the webbing of a particular setting which, in turn, shapes the way the ideas are expressed” (p. 122). Webbing in this sense means “the presence of a structure that learners can draw up and reconstruct for support – in ways that they can choose as appropriate for their struggle to construct meaning for some mathematics (Noss & Hoyles, 1996, p. 108). Hershkowitz, Schwarz, and Dreyfus (2001) introduced the notion of abstraction in context that they presented as “an activity of vertically reorganizing previously constructed mathematics into new mathematical structure” (p. 202). They identify abstraction in context with what Treffers (1987) described as ‘vertical mathematization’ and propose entire mathematical activity as the unity of analysis. These contributions demonstrate that research on abstraction in knowing and learning mathematics has made significant progress in taking account of the context-sensitivity of knowledge. Several other contributions shape the territory in mathematics education research on abstraction. Scheiner (2016) proposed a distinction between two forms of abstraction, namely abstraction on actions and abstraction on objects. This distinction has been further refined in Scheiner and Pinto (2014) arguing that the focus of attention of each form of abstraction takes place on physical objects (referring to the real world) or mental objects (referring to the thought world) (see Fig. 1). abstraction on actions abstraction on objectsion on actions abstraction on objects