This paper represents a first step in the development of a framework which can be used to guide research into the relationships between problem posing and problem solving in school mathematics. A framework is presented in which the types of problem-posing situations used in mathematics classrooms are classified. Three types of structures for problem-posing situations are identified--free, semi-structured and structured--and these can be used to generate problem-posing situations for mathematics classrooms. Emphasis is given to Krutetskii's problem-solving categories, and to how these can be extended to the domain of problem posing.

Introduction

After over a decade of studies which have focused on problem solving, researchers have slowly begun to realise that developing the ability to pose mathematics problems is at least as important, educationally, as developing the ability to solve them. Kilpatrick (1987) and Silver (1993) are among many mathematics educators who have suggested that the incorporation of problem-solving and problem-posing situations into mathematics classrooms could have a positive impact on students' mathematical thinking.

However, research into the potential of problem posing as an important strategy for the development of students' understanding of mathematics has been hindered by the absence of a framework which links problem solving, problem posing and mathematics curricula. Before the effects of problem posing and its implication for the teaching and learning of mathematics can be adequately researched, such a framework needs to be developed and refined in the light of data gained from its application in the classroom.

Duncker (1945) described problem posing as the generation of a new problem or the reformulation of a given problem. In this paper mathematical problem posing will be defined as the process by which, on the basis of mathematical experience, students construct personal interpretations of concrete situations and formulate them as meaningful mathematical problems.

The potential of using problem-posing activities in the teaching and learning of mathematics has been explored from various perspectives:

1. As a way of extending students' understanding of important mathematical ideas (Ellerton 1986; Pegg & Davey, 1991);

2. As a means for improving students' skills in problem solving (Hashimoto & Swada, 1984; Shimada, 1977; Silver & Cai, 1993);

3. As a way of investigating students' difficulties and mathematical abilities ( Ellerton, 1986; Krutetskii, 1976);

4. As a way of preparing students to be intelligent users of mathematics in their everyday lives (Blum & Niss, 1991; Writz & Kahn, 1982); and

5. As a way of linking students' own interests with their mathematical education (Bush & Fiala, 1986).

A model for research into students' problem posing in a problem-solving environment can be developed by looking at Krutetskii's (1976) analysis of children's mathematical abilities from a problem-posing perspective. In this paper an overview of the types of problem-posing situations which have been used in a classroom with mathematically gifted children will be presented.

A Framework