PROBLEM SOLVING: ITS ASSIMILATION TO THE TEACHER'S PERSPECTIVE Paul Ernest University of Exeter INTRODUCTION It could be said that there is a problem-solving bandwagon rolling. Many influential reports published in the USA, UK and elsewhere, such as the Agenda for Action (NCTM, 1980), the Cockcroft Report (DES, 1982), the Standards (NCTM, 1989) and the National Curriculum in Mathematics (NCC, 1989), all strongly endorse a problem solving approach to school mathematics. These recommendations have been made for over a decade -- three decades, if we include the 'discovery learning' of the 1960s. There are powerful arguments behind these endorsements, which I will not rehearse here. But the fact is that most mathematics teaching remains routine and 'instrumental'. More often than not, children are given a method for carrying out a type of task, and then many graded exercises to practice and reinforce the method. Each task has a unique correct answer. Of course such procedures do generate some 'relational' and well as 'instrumental' understanding on the part of the learner, and perhaps some strategic skills. But the primary focus of such exercises is the successful acquisition and deployment of procedures, and the acquisition of relational understanding or strategic skills is incidental. So what is going on? Why the disparity between the problem solving recommendations and the routine and convergent nature of much of school mathematics? There are many reasons, including institutional resistance to change, vested interests behind the status quo, individuals' resistance to change, be they teachers, learners, parents or others, and so on. In this paper I want to pick out one strand to explore, which in my view has received insufficient attention. That is the assimilation of problem solving to the teacher's perspective, by which I mean the teacher's mathematics-related belief-system. Teachers have different beliefs about the nature of mathematics and its teaching and learning, which powerfully affect their classroom practices. In some form or other, this relationship is now widely accepted, and I elaborate on it below. But one unremarked consequence of this, I wish to claim, is that problem solving is understood differently by different teachers, in accordance with their beliefs. A demonstration of the lack of unique meaning and the diversity of interpretations given to the term 'problem-solving' by teachers and others might go some way towards explaining how widespread espousal of problem-solving can co-exist with its widespread rejection in practice. THE MEANINGS OF PROBLEM SOLVING Problem solving has been a widespread part of the rhetoric of British mathematics education since Cockcroft (1982). Worldwide, problem