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In this article, the authors share a European perspective on teaching linear equations in Finland, Flanders, and Hungary. They begin with an assertion that it is through observation, in its broadest sense, and opportunities for participation, that people assimilate themselves into the culture in which they are raised. They discuss three approaches to mathematics teaching and reflect on the one characteristic common to these approaches. They compare that commonality with what they have seen in other European countries. This interesting commonality is that trainees have always begun their formal treatment with arithmetic equations; equations with the unknown on one side that they know their students can solve mentally.

This paper addresses four questions concerning the influence of culture on mathematics teachers' professional practice. Firstly, drawing on categorical data yielded by the application of low inference coding schedule to video recordings of sequences of lessons taught by case study teachers on four common topics in England, Flanders, Hungary and Spain, we undertook an exploratory factor analysis to examine the ways in which such coded variables interact. This process yielded five factors, each of which was interpretable against the literature and highlighted the extent to which dichotomisations of mathematics teaching as reform or traditional are not necessarily helpful, not least because all project teachers exhibited characteristics of both. Secondly, factors scores were analysed by nationality to reveal culturally located practices resonant with the available literature. Thirdly, cluster analyses yielded four well-defined cross-cultural clusters of episodes, each indicative of particular didactical perspectives that appeared to challenge the exclusivity of these culturally located practices. Finally, the key methodological finding was that the manner in which data are analysed influences greatly the outcomes of comparative mathematics research. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we compare how three teachers, one from each of Finland, Flanders and Hungary, introduce linear equations to grade 8 students. Five successive lessons were videotaped and analysed qualitatively to determine how teachers, each of whom was defined against local criteria as effective, addressed various literature-derived equations-related problems. The analyses showed all four sequences passing through four phases that we have called definition, activation, exposition and consolidation. However, within each phase were similarities and differences. For example, all three constructed their exposition around algebraic equations and, in so doing, addressed concerns relating to students’ procedural perspectives on the equals sign. All three teachers invoked the balance as an embodiment for teaching solution strategies to algebraic equations, confident that the failure of intuitive strategies necessitated a didactical intervention. Major differences lay in the extent to which the balance was sustained and teachers’ variable use of realistic word problems. [Copyright &y& Elsevier]