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A further reason why a problem-solving approach is valuable is as an aesthetic form. Problem solving allows the student to experience a range of emotions associated with various stages in the solution process. Schoenfeld also suggested that a good problem should be one which can be extended to lead to mathematical explorations and generalisations.
Through a problem-solving approach, this aspect of mathematics can be developed.
Presenting a problem and developing the skills needed to solve that problem is more motivational than teaching the skills without a context.
Problem solving is, however, more than a vehicle for teaching and reinforcing mathematical knowledge and helping to meet everyday challenges.
It is also a skill which can enhance logical reasoning.
Specific characteristics of a problem-solving approach include: My early problem-solving courses focused on problems amenable to solutions by Polya-type heuristics: draw a diagram, examine special cases or analogies, specialize, generalize, and so on.
Over the years the courses evolved to the point where they focused less on heuristics per se and more on introducing students to fundamental ideas: the importance of mathematical reasoning and proof..., for example, and of sustained mathematical investigations (where my problems served as starting points for serious explorations, rather than tasks to be completed).
Mathematicians who successfully solve problems say that the experience of having done so contributes to an appreciation for the 'power and beauty of mathematics' (NCTM, 1989, p.77), the "joy of banging your head against a mathematical wall, and then discovering that there might be ways of either going around or over that wall" (Olkin and Schoenfeld, 1994, p.43). 'Constructing meaningful understanding of mathematics content', in Aichele, D. (Eds.) Professional Development for Teachers of Mathematics , pp.
They also speak of the willingness or even desire to engage with a task for a length of time which causes the task to cease being a 'puzzle' and allows it to become a problem.
In the past decade it has been suggested that problem-solving techniques can be made available most effectively through making problem solving the focus of the mathematics curriculum.
Although mathematical problems have traditionally been a part of the mathematics curriculum, it has been only comparatively recently that problem solving has come to be regarded as an important medium for teaching and learning mathematics (Stanic and Kilpatrick, 1989).