*As was pointed out earlier, standard mathematics, with the emphasis on the acquisition of knowledge, does not necessarily cater for these needs. Resnick (1987) described the discrepancies which exist between the algorithmic approaches taught in schools and the 'invented' strategies which most people use in the workforce in order to solve practical problems which do not always fit neatly into a taught algorithm. According to Resnick (1987) a problem-solving approach contributes to the practical use of mathematics by helping people to develop the facility to be adaptable when, for instance, technology breaks down. (Eds.) Professional Development for Teachers of Mathematics , pp. An Agenda for Action: Recommendations for School Mathematics of the 1980s, Reston, Virginia: NCTM. *

Individuals can no longer function optimally in society by just knowing the rules to follow to obtain a correct answer.

They also need to be able to decide through a process of logical deduction what algorithm, if any, a situation requires, and sometimes need to be able to develop their own rules in a situation where an algorithm cannot be directly applied.

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).

'If education fails to contribute to the development of the intelligence, it is obviously incomplete.

## Problem Solving With Solution In Math Problem Solving Activities Teens

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). As the emphasis has shifted from teaching problem solving to teaching via problem solving (Lester, Masingila, Mau, Lambdin, dos Santon and Raymond, 1994), many writers have attempted to clarify what is meant by a problem-solving approach to teaching mathematics. The focus is on teaching mathematical topics through problem-solving contexts and enquiry-oriented environments which are characterised by the teacher 'helping students construct a deep understanding of mathematical ideas and processes by engaging them in doing mathematics: creating, conjecturing, exploring, testing, and verifying' (Lester et al., 1994, p.154). 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. 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. 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. 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.

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