Mathematical Abstraction through Scaffolding.
Mehmet Fatih Özmantar, Tom Roper · Proceedings of the ... PME Conference · 2004
ion is a difficult activity that can sometimes be beyond students’ unassisted efforts, in which case supportive intervention of a scaffolder through several means of assistance is observed to help the students achieve the abstraction. The issue of abstraction has attracted the attention of many educators (Dienes, 1963; Piaget, 1970; Skemp, 1986). Purely cognitive views see abstraction as ascending from ‘concrete’ to ‘abstract’, e.g. “the extraction of what is common to a number of different situations” (Dienes, 1963, p.57). Criticism of this view of abstraction comes from an epistemological point of view which recognises that contextual and social factors are crucial to knowledge acquisition (see van Oers, 2001). Empirical studies of abstraction in context are a relatively recent phenomenon and include Noss and Hoyles’ (1996) concept of situated abstraction and Hershkowitz, Schwarz and Dreyfus’ (2001) activity-theoretic model of abstraction. This paper works within the Hershkowitz et al.’s model. Hershkowitz et al.’s (2001) model was inspired by Davydov’s (1990) epistemological theory pointing a dialectical connection between abstract and concrete. They provide an operational definition of abstraction as “an activity of vertically reorganising previously constructed mathematics into a new mathematical structure.” The new structure is the product of three epistemic actions: recognising, building-with, and constructing (‘RBC theory of abstraction’ hereafter). Recognising a familiar structure occurs when a student realises that a structure is inherent in a given mathematical situation. Building-with consists of combining existing artefacts in order to meet a goal. Constructing consists of assembling knowledge artefacts to produce a new structure. These actions are dynamically nested in such a way that building-with includes recognising, and constructing includes both recognising and building-with. In empirical studies of abstraction researchers claim to gain insight into students’ abstraction processes via the help of a knowledgeable agent e.g. the researcher/ interviewer. Hershkowitz et al. (2001), for example, note that the interviewer in their study aims to induce the student to reflect on what she is doing so that she might progress beyond the point that she would have reached without the interviewer. Others argue that the successful completion of an abstraction process is contingent upon providing the student with ‘hinting’ (Ohlsson and Regan, 2001) and ‘shifting the focus of activities’ (van Oers, 2001). All three sets of researchers effectively argue that abstraction is not an easy process which may be beyond the learners’ unassisted efforts. This has links with the theoretical concept of ‘scaffolding’.