The objective of this study is to establish the strategies which are used in spatial tasks and to determine whether and how the visualisation skills used in the performance of such tasks can be taught. The primary focus is on measuring the level of complexity when comparing different knot shapes and on analysing the subjects' verbal and visual strategies. To address these issues, quantitative and qualitative studies are carried out, firstly to identify the factors which affect complexity when comparing knots and, secondly, to identify the strategies that are most effective. The first phase of research measures two complexity indicators: the time taken to perform the task and the accuracy of response when comparing pairs of knots at varying orientations in order to determine whether the two knots are the same or different. The endogenous factors associated with the level of difficulty are knot shape, crossing number, rotation and pair type, and the exogenous factors considered in the statistical analysis are gender and educational background. Six different knot shapes are used and the results show that some knot shapes are processed more efficiently than others, a finding similar to that in psychology where images which have a characteristic 'foundation part' are more easily stored and accessed in the brain. The results also reveal that there are differences in time and accuracy for topologically different pair types and that knot rotation causes an increase in decision times and error rate. During the experiment, subjects show a significant improvement in their ability to perform the knot tasks, indicating that imagery and spatial skills are enhanced as a result. In the second phase of research, where subjects explain their changes of strategy whilst viewing a recording of the experiment, evidence is provided concerning the way in which the performance of the knot tasks improves spatial and visualisation skills. It is shown how the nature and deformability of knots allow strategies other than the self-evident one of mental rotation, and the research therefore establishes alternative methods that may be employed to develop imagery in mathematics.