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.