Many children have difficulties learning mathematics, and the consequences of poor mathematical skills are very far reaching. Studies examining the reasons why children struggle to learn mathematics are scarce, particularly in comparison to studies examining reading difficulties. The studies reported in this thesis attempted to provide insights into the cognitive limitations that may lead some children to have difficulties learning mathematics, re-examining some of the cognitive deficits already thought to be associated with mathematical difficulties, as well as providing the starting point for new lines of enquiry. Five main studies are reported. Four of these studies examined a range of cognitive skills and identify a number of fundamental cognitive mechanisms as playing a role in children's mathematical skills, these being a slowness in the speed of processing information, poor control of executive functioning, evidenced through difficulty switching strategies and poor self-regulation of actions, and a delay in the automatization of basic arithmetic facts. The final study aimed to investigate the implications of these recognised cognitive difficulties in the teaching of mathematics, and explored the use of two different teaching strategies, rote learning of basic arithmetic facts and a discussion method to allow alternative methods of solution to be learned, both of which attempted to overcome some of these cognitive limitations. Rote learning was found to be an effective device to improve performance in different areas of mathematical skill. The implications of this research and the foundations for future research are also discussed.