Problem-posing situations in mathematics classrooms can be classified on the basis of characteristics and certain structural features of the situations themselves. Central to this paper is the thesis that any problem-posing situation can be classified as free, semi-structured or structured.

Free Problem-posing Situations

In a free problem-posing situation the problem is not given. Students are simply asked to pose a problem. Directions such as: "Make up a difficult problem," or "Construct a problem suitable for a mathematics competition (or a test)," or, simply, "Make up a problem you like," can help students to reflect on and to mathematise their previous experiences. Free problem-posing environments often generate problems which derive from out-of-school experiences and contain surplus or incomplete information.

In real-life situations, individuals are often moved to construct idiosyncratic problems. For example, a child who would like to own a pair of roller blades might formulate a method by which she can save enough money to buy a pair. She might ask herself "How many hours of baby-sitting would I need to do to earn enough," and "What might I sell to raise the extra I need?" Finally, she may ponder "How long would it be before I can buy the roller blades?"

Semi-structured Problem-posing Situations

In semi-structured problem-posing situations students are given an open situation and are invited to explore it using knowledge, skills, concepts and relationships from their previous mathematical experiences. Open-ended problems of this kind are often termed "mathematical investigations." According to Kissane (1988), a critical, defining feature of an investigation is that the student is responsible for devising, refining, and pursuing the questions.

Structured Problem-posing Situations

In a structured problem-posing situation, a specific problem or problem solution is given, and the problem-posing task is to develop new problems which derive from the given problem or problem solution (see, for example, Butts, 1980).

Polya (1957) has mentioned three approaches for constructing a new problem from a proposed problem: "Firstly, keep the unknown and change the rest (data and the condition); or secondly keep the data and change the rest (the unknown and the condition); or thirdly change both the unknown and the data" (p. 78). Brown and Walter (1990, 1993), who also designed an instructional problem-formulating approach based on the posing of new problems from already solved problems, have also recommended varying the conditions or goals of given problems. This reformulation approach appears to be the most popular method for introducing structured problem posing activities in mathematics classrooms.

Polya (1957) recommended asking students to make up a problem with the same method or solution. Kilpatrick (1987) has argued that, when students attempt to solve problems, there are two phases during which new problems can be created. "As a mathematical model is being constructed for a problem, the solver can intentionally change some or all of the problem conditions to see what new problem might result. After a problem has been solved, the solver can look back to see how the solution might be affected by various modifications in the problem" (p. 127).

From the above discussion, the author would contend that the following three basic principles can be applied by educators wishing to develop quality structured problem-posing situations in mathematics classrooms:

1. Problem-posing situations should correspond to, and arise out of, pupils' classroom mathematics activities;

2. Problem-posing situations should correspond to pupils' problem-solving processes;

3. Problem-posing situations can be generated from textbook problems, by modifying and reshaping the language and task characteristics.

Table 1, below, summarises the situations which might be associated with free, semi-structured and structured problem-posing situations.

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