Applying Lakatos' theory to the theory of mathematical problem solving

In this paper, the relation between Lakatos' theory and issues about mathematics education — especially issues about mathematical problem solving — is reinvestigated by paying attention to Lakatos' methodology of a scientific research programme. By comparing the same findings about mathematical problem solving with the discussion in Lakatos' theory — e.g. research programmes' hard cores, their negative and positive heuristics, and their goals — we establish the correspondence between research programmes and solver's structures of a problem situation, i.e. structures given by a solver to a problem situation. After establishing this, the implications of Lakatos' theory, i.e. the nature of selection from competing programmes and the social origins of the cores of programmes, are applied to the discussion about mathematical problem-solving, with indications of the related evidence in the theory of mathematical problem solving which seems to support the application of those implications. Such an application leads to one view of mathematical problem solving, which reflects the irrational nature and social aspects of problem-solving activities, both in solving problems and in selecting better solutions.