An investigation of the formation of mathematical abstractions through scaffolding
This study takes an activity-theoretic approach to abstraction in context recently proposed by Hershkowitz, Schwarz and Dreyfus (2001, HSD hereafter). Key to HSD's theory of abstraction is the construction of new mathematical knowledge and consolidation of it. In this connection, this study aims to investigate three particular issues: (1) the construction of mathematical knowledge through scaffolding, (2) the nature of the consolidation process and (3) the validity ofHSD's abstraction theory. In order to investigate these issues, a qualitative research design methodology with explanatory and exploratory inquiry purposes was taken. This study employed multiple case study strategy with the purpose of literal and theoretical replications. A number of cases were designed with students working as pairs and individuals such that some of the students worked with the scaffolded help and others without. All participants worked on four days over four sequential tasks connected with the graphs of absolute value functions. Tasks were applied in paper-andpencil format. The data for this study was composed of the participant's written works and audio records of the sessions. In relation to the first issue, analysing the students' verbal data suggests certain causative relationships between the scaffolder's interventions and the students' developing constructions. It is also observed that the scaffolder's interventions mediate the students' constructions. Analysis of the data further suggests that construction through scaffolding is a subtle and intricate phenomenon which involves a complex set of social, cultural, historical, contextual and semiotic issues. It is argued, with examples, that scaffolded discourse involves many dynamics such as value judgements, individuals' personal histories, common cultural practices, individuals' emergent goals, voices of absent others and certain patterns of interaction. Regarding the second issue, the data suggest that newly formed constructions are fragile entities and in need of consolidation. In the course of consolidation, it is observed that earlier constructions are reconstructed, used in a flexible manner and expressed confidently with general mathematical statements. These observations lead to the argument that an abstraction is a consolidated construction that can be used to create new constructions. With regard to the final issue, on the basis of the students' verbal data, this study provides a critical evaluation of HSD's theory of abstraction by focusing on three key dimensions which characterise it: its epistemological and sociocultural principles, epistemic actions and genesis of an abstraction. Throughout this evaluation some clarifications and amendments are proposed to this theory.