Abstract
D.Phil. (Mathematics Education)
In tertiary education, a statement like ‘Low graduation rates prevail around the world’ is common knowledge since the 1940s, and therefore one does not need any longer to mention references. The factors that contribute to it though, are innumerable. One of those factors is the ability of a student to solve problems. Problem solving has been accepted as a prerequisite for lifelong learning by many governments and it is enshrined in their educational policies. However, problem solving can be associated with academic performance (mastery of content knowledge being a main contributor) as well as application/transfer of content knowledge. Critical thinking on the other hand is embedded in problem solving, acquisition of knowledge and application. Then an investigation into the relationships between all these constructs is warranted.
This research aimed at shedding some or more light into this proverbial problem. Problem solving is equated by some authors to learning. Learning while solving problems and solving problems result in learning. Almost all theorists see problem solving as a process and be one of the products of learning. This research concluded that problem solving is a product of its own as a result of a number of complex cognitive processes. The simple argument is: If a problem solver cannot solve a problem successfully then no product is produced by those cognitive processes. In actual fact, the possibility of the existence of misconceptions could be one of the reasons for the failure of solving the problem. If that is true, then the statement: ‘we should be diagnosing rather than teaching’ could be valid.
Furthermore, teaching problem solving as a process gives rise for it to be treated as an ‘algorithm’ by students which they try to memorise without having a conceptual understanding of the problem. However if it is treated as a product the students will be encouraged to think of the various cognitive processes that are necessary to solve the problem. This research concluded that cognitive processes such as critical thinking, acquisition of (mathematical) knowledge and application thereof, can lead to a product which was guided by ‘quality control processes’. Therefore problem solving in this research is not explicitly expressed but implicitly. As a result ‘successful problem solving’, the product, is closely associated with academic achievement.