ABSTRACT. This paper draws on videotapes of mathematics lessons prepared and conducted by pre-service elementary teachers towards the end of their initial training at one university. The aim was to locate ways in which they drew on their knowledge of mathematics and mathematics pedagogy in their teaching. A grounded approach to data analysis led to the identification of a 'knowledge quartet', with four broad dimensions, or 'units', through which mathematics-related knowledge of these beginning teachers could be observed in practice. We term the four units: foundation, transformation, connection and contingency. This paper describes how each of these units is characterised and analyses one of the videotaped lessons, showing how each dimension of the quartet can be identified in the lesson. We claim that the quartet can be used as a framework for lesson observation and for mathematics teaching development.

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Our research in the last decade has been into classroom situations that we perceive to make demands on mathematics teachers' disciplinary knowledge of content and pedagogy. Amongst the most visible of such situations are those that we describe as ‘contingent’, in which a teacher is faced with some unexpected event, and challenged to deviate from their planned agenda for the lesson. Our findings and the associated grounded theories have been open to enhancement and revision as new classroom data has been gathered. In this article, we propose a classification of the origins of contingent classroom episodes: namely the students; the teacher him/herself; and pedagogical tools and resources. This classification extends and elaborates our original conception of ‘contingent’, in response to more recent data

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