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Through a problem-solving approach, this aspect of mathematics can be developed.
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.
'If education fails to contribute to the development of the intelligence, it is obviously incomplete.
Yet intelligence is essentially the ability to solve problems: everyday problems, personal problems ... Modern definitions of intelligence (Gardner, 1985) talk about practical intelligence which enables 'the individual to resolve genuine problems or difficulties that he or she encounters' (p.60) and also encourages the individual to find or create problems 'thereby laying the groundwork for the acquisition of new knowledge' (p.85).
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.
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).
As she says, most people have developed 'rules of thumb' for calculating, for example, quantities, discounts or the amount of change they should give, and these rarely involve standard algorithms.
Training in problem-solving techniques equips people more readily with the ability to adapt to such situations.