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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.
Let us consider how problem solving is a useful medium for each of these.
It has already been pointed out that mathematics is an essential discipline because of its practical role to the individual and society.
Schoenfeld also suggested that a good problem should be one which can be extended to lead to mathematical explorations and generalisations.
He described three characteristics of mathematical thinking: Problem solving is an important component of mathematics education because it is the single vehicle which seems to be able to achieve at school level all three of the values of mathematics listed at the outset of this article: functional, logical and aesthetic.
The National Council of Teachers of Mathematics (NCTM, 1980) recommended that problem solving be the focus of mathematics teaching because, they say, it encompasses skills and functions which are an important part of everyday life.
Furthermore it can help people to adapt to changes and unexpected problems in their careers and other aspects of their lives.
Such motivation gives problem solving special value as a vehicle for learning new concepts and skills or the reinforcement of skills already acquired (Stanic and Kilpatrick, 1989, NCTM, 1989).
Approaching mathematics through problem solving can create a context which simulates real life and therefore justifies the mathematics rather than treating it as an end in itself.
However, although it is this engagement which initially motivates the solver to pursue a problem, it is still necessary for certain techniques to be available for the involvement to continue successfully. Report of the Committee of Inquiry into the Teaching of Mathematics in Schools, London: Her Majesty's Stationery Office.
Hence more needs to be understood about what these techniques are and how they can best be made available.