The importance of problem solving activities is emphasised in the mathematics curriculum all over the world. However, still not enough is known about how people can best be taught problem-solving skills. Moreover, there are many difficulties associated with teaching people how to succeed at problem solving (Taplin, 1998). Answers to these questions are extremely important for developing further the theoretical framework of problem solving, and have practical implications in work with gifted mathematics students. Most high-profile students regularly participate in numerous mathematical competitions and, for them to achieve the best results, their training should be grounded on a comprehensive theoretical base. Why are just a few students able to be successful in mathematical competitions, while many others fail with this challenge? Does this mean that just a handful of mathematically gifted students can achieve a high-level of mathematical thinking? Or, is there scope for others to be more successful in problem solving?

In this lecture I will consider how the development of conceptual understanding may lead to the design and realisation of problem solving activities, which do impact positively on students' performance in mathematical competitions. In analysing the role of conceptual understanding, I focus in particular on the influence of mathematical reading, carried out by students, and mathematical explanations, provided by teachers. Mathematical reading provides a challenge to understand a text and work up a strategy resolving a given task (Mamona-Downs & Downs, 2005). Mathematical explanations are used to highlight a more general approach that can be applied and elaborated beyond a given task; they allow the reorganisation of problem solving activities according to explanatory principles. Examples of mathematical reading and explanation will be given and their practical implications discussed. The key practical outcome is the development of the learner's skill to recognise similarities amongst problems and links between them, which is one of the most influential factors in problem solving.

Finally I will talk about the importance of creating an opportunity for learners to experience the excitement of mathematical constructions and the power of mathematical knowledge. Mathematics, in both teaching and curriculum, must be made more enticing (Watson, 2008), rather than being further simplified. Problem solving is a vehicle for promoting and developing this excitement and power.