Expressing generality : young children, mathematics and Logo
This thesis looks at young children's attempts to express generality within specific learning situations with the aid of carefully designed tools, and at how, given appropriate means of expression, they are able to justify their generalisations. The literature on representation and abstraction leads to a focus on construction of meaning through pupils' concretising of mathematical objects by means of the development of representations and interconnections between representations of those objects. The study involved eight pairs often and eleven year old children. I devised a series of tasks centred on the creation of simple "function machines" expressed in the Logo programming language. The children's work involved the construction and empirical testing of these functions within a game-like situation. Part of the game involved verbally justifying the validity of the Logo procedures to a partner and to the Researcher. These activities provided a window onto children's construction of meaning. Analysis of the data revealed that within the specific learning situation designed for the study: children were able to make formalised generalisations of mathematical relationships, often webbed by "semi-generalisation"; the expressive powers of the symbolism achieved a more functional role by the symbols' association with a history of specific numerical examples; children constructed situated abstractions for the justification of generality using "generic structuring" and "naturalised formalism" as powerful forms of webbing; and the apparent "rift" between empirical and deductive starting points for generalisation, justification and proving activities appeared less clear than the literature suggests.