UBC Theses and Dissertations

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UBC Theses and Dissertations

The grammar of symbolic elementary algebra Kirshner, David


In the practise of algebra education it is generally assumed that the rules for manipulating symbols are explicitly presented by text and teacher, and that acquisition of skill depends upon successful apprehension and application of these rules. Research into the psychology of algebra generally starts from the assumption that algebraic rules are consciously accessible and rationally employed. The view adopted here is that algebraic symbol skill is based upon procedural rules which are acquired informally (and often unconsciously) through interaction with algebraic symbols and which may be only peripherally related to the rules presented in instruction. The principle purpose of the present research is to describe the procedural knowledge which underlies algebraic symbol skill. Knowledge of algebra is viewed as of-a-kind with the highly structured yet unconscious systems of rules which underlie natural-language competence. Formal methods of generative transformational linguistics are adapted for analysis of algebra. The model of algebraic symbol skill is captured in a 'grammar' which details the various components of skill. In several instances alternative formulations are offered which equally well fulfill the formal requirements of the grammar. For some of these, techniques of psycholinguistics are used to guide selection on the basis of psychological considerations. Many questions remain in need of further elaboration and resolution.

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