Mathematics teaching in the UK has undergone several major externally imposed changes during the last decade. Current practices display a range of epistemological and pedagogical assumptions and behaviours, depending on teachers' interpretation of, adoption of and belief in current statutory requirements for teaching and assessment. This paper examines in detail differences within the informal assessment practices of 30 UK mathematics teachers. It is found that these illustrate several of the paradigmatic differences that permeate studies of human behaviour on a grander scale. Since informal assessment decisions can lead directly or indirectly to differentiated access to the curriculum and high-stakes grading, the use of teacher assessment as a focus for examining differences illuminates the possible inequities which might arise for pupils. Examination of differences within one system and one society gives information about effects of different educational practices which, were they to show up between societies, might be attributed to other social and cultural factors.
The recent publication of the TIMSS report (1997) has generated much interest in international comparisons of the mathematical performance of school children. Many attempts have been made to make sense of different outcomes relative to features of national culture, such as predominant teaching styles, social structures, educational intentions and so on (Jaworski and Phillips, 1999). In this paper I will look closely at a subgroup of differences of practice within one highly structured national system, that of the UK National Curriculum (NC), in order to show a range which cannot be explained solely by national features. These differences reveal a collection of paradigmatic conflicts within one national culture, expressed as variations in classroom cultures.
In the context of educational research Gage (1989) suggests that 'paradigm differences do not require paradigm conflict'. He shows how it should be possible for three apparently unreconcilable perspectives, which I shall roughly classify positivist, relativist and emancipatory, to contribute three compatible forms of knowledge about teaching, learning and schools. The kinds of questions one can pose within each paradigm, and the ways one might answer, are different but the ultimate purpose of improving education is the same. However, Brown (1993) describes conflicts that arise when these