Analogical Reasoning and the Development of Algebraic Abstraction

This paper presents a theory of the development of algebraic abstraction which extends Sfard's (1991, 1994a,b) and Mason's (1982, 1989) ideas on the learner's progress from operational or process-oriented thinking to the abstract or structural perspective. The theory incorporates a process of analogical reasoning to account for the means by which the learner might construct expressions of generality and subsequently manipulate them as mathematical objects. Such reasoning entails similarity comparisons in which a mapping is made between the corresponding relational properties of algebraic examples. These comparisons may firstly entail "unpacking the relations" in the examples in order to highlight the structural commonalities. The common relational structure is subsequently extracted to form a knowledge basis, namely, the construction of a mental model or representation that expresses the observed generalisation. The theory is applied to an analysis of secondary school students' approaches to classifying a set of complex equations. A student who appeared capable of algebraic abstraction within the domain of the task is contrasted with two students who were at a "pseudostructural" stage, where their focus on syntactic surface structures prevented them from forming the relational mappings needed for the construction of generalised models of the equation types.