Mathematical problem solving

Problem solving has different meanings to different people and it is difficult to reach a common understanding of the concept of problem solving. Broadly, to solve a problem means finding a way where no way is known, finding a way out of difficulty, finding a way around an obstacle, or attaining a desired end that is not immediately attainable by appropriate means (Hatfield, Edwards, Bitter & Morrow, 2000:91). According to Johnson and Rising (1967:104), problem solving means “finding an appropriate response to a situation which is unique and novel to the problem solver”. According to Silver (1987:40), problem solving means the application of one’s knowledge to tasks that may be well structured or poorly structured, familiar or unfamiliar, simple or complex. Problem solving can also be perceived as the process of getting from givens to goals when the path is not obvious (Lesh & Doerr, 2003:31).

Not all learners in a class may view what is being taught as a problem. Orton and Frobisher (2000:25) put forward that what is a problem to one learner may be an exercise for another learner. For instance, those who have little understanding of a situation may view any mathematical idea arising from an activity associated with the situation as a problem. This means that learners who have already met a situation before and have become reasonably familiar with the different aspects of the mathematics of the activity will view their work as a repetitive exercise.

For example, in an activity that requires learners to solve for x in2x 3using a calculator and leaving the answer in decimal form, a learner used to solving exponential equations using logarithms is likely to obtain the answer of 1,584962501. Such a learner does not see exponential equations having different bases as a problem. However, another learner who is at early stages of solving exponential equations can regard the equation2x 3as a problem. Such a learner will employ many strategies, methods and processes to arrive at a solution. One learner may use trial and error methods, substituting x with 0,5; 1; 1,5 and 2

using a calculator and estimate the answer as lying between 1 and 2 instead of arriving at the exact solution of 1,584962501. However, a less advanced learner may say we cannot solve the equation, as the bases are not the same. Stacy (2005:342) states that successful mathematical problem solving depends on deep mathematical knowledge, reasoning abilities,

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communication skills, abilities to work with others and personal attributes such as persistence, organisation and confidence.

Mathematical problems consist of three types of information, namely information regarding ‘givens’, information regarding ‘operations’ and information concerning ‘goals’ (Wickelgren, 1995:10). Givens refers to the set of expressions that we accept as being present at the onset of the problem statement, operations refers to the actions one need to perform on the givens, while goals implies a terminal expression one wishes to cause to exist in the problem situation (Wickelgren, 1995:13). In the problem such as “What constant force will cause a mass of 3 kilograms to achieve a speed of 30 metres per second in 6 seconds, starting from rest?” the givens are 3 kilograms, 30 metres and 6 seconds. In the proof problems, the rule of inference that constitutes the allowable operation is the property that if A A' and BB', then ABA' B '. For example, in the problem 2x719, one can regard the goal expression as being of the form x ---, where the correct number is to be found in order to fill in the blank in the goal expression.

Pollack (1969:397) cautions mathematics teachers to avoid problems that look like applications to mathematics when in actual fact they are not. He gives the following scenario as an example of such problems:

A bee and a lump of sugar are located at different points inside a triangle. The bee wishes to reach the lump of sugar, while travelling a minimum distance, under the requirement that it must touch all three sides of the triangle before coming to the sugar. What is the shortest path?

One advantage of such problems may be that they bring a smile to a weary learner or distract him or her momentarily from a ‘dreary lesson’ and divert the imagination to a more pleasant exercise (Pollack, 1969:398). Such teachers act as if the learners’ interests must be enhanced by including problems or situations coming from outside the mathematics classroom (Stigler & Hiebert, 1998:2).

Orton and Frobisher (2000:27) mention three categories of mathematical problems, namely routine problems, environmental problems and process problems. Routine problems are problems that use knowledge and techniques already acquired by a learner in a narrow and

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systematic context. Environmental problems, also called ‘real-world’ or ‘real-life’ problems, are problems set in contexts that represent the practical or real world. Process problems are set in a mathematics context in contrast to real problems. These types of problems concentrate on the mathematics itself and on the mathematics thinking processes for arriving at the solution. The Pollack problem stated above can fall under the category of environmental problems, as the problem relate to real-life problems even though the problem is not purely mathematical.