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It is widely recognised that generality is at the heart of the learning and teaching of mathematics. Motivated by a desire to understand what it is about generality which presents such an obstacle for so many students, this study examines the variety and complexity of ways in which generality is expressed in mathematics classrooms. Systematic reflection on my own experience of teaching over a year revealed a wide range of types of generalisation taking place in mathematics classrooms. The main study then analyses transcripts of fifty-two lessons taught by six teachers teaching at least four hundred students, sampled over a period of two months. The focus is on 'ordinary' lessons where expression of generality is not the main objective. Infonned by the literature, observation notes and student work, a framework is developed with five categories used to distinguish between types of generalisations, which emerge from the transcribed data. These categories are: the object of generalisation, its presumed longevity of relevance, its justification, its origin and the awareness being promoted. Having established the Ubiquitous richness and complexity of expression of generality in mathematics classrooms, the study looks in closer detail at the expression of generality pertinent to mathematical procedures and to mathematical concepts. The study uses the framework, and draws on second language education literature, to re-examine the fifty-two main study lessons. This analysis highlights the complexity of expressing generality through natural language, and suggests that natural language exhibits many of the pitfalls and ambiguities of algebraic expression. Further, it suggests that algebraic notation might offer a clearer means of expressing generality in many cases. The framework developed for considering characteristics of expressions of generality is then applied to the researcher's own classroom, demonstrating how awareness of ways in which generality is expressed can inform pedagogic choices as well as provide a structure for reflection on practice. |
