The research literature on children's understanding of randomness has developed\ud considerably in recent decades, notably due to the key contributions of researchers such as\ud Piaget and Inhelder (1975) and Tversky and Kahneman (1983). Yet the research has paid\ud rather scant attention to the tools that people have available for expressing ideas about\ud randomness, fairness and more generally, probability. In contrast, work within the\ud paradigm of 'constructionism' makes the explicit claim that by using tools that are\ud specially designed for expressing concepts of randomness and chance, people may be\ud better able to express ideas that can seldom be predicted by cognitive analysis based on,\ud say, misconceptions or thinking stages that fail to take sufficient account of too] mediation.\ud This study investigated the nature of young children's expressions of random events.\ud Specifically the aims of the study were:\ud E iteratively to design and evaluate a tool-based game to afford children between the\ud ages of 5 1/2and 8 opportunities to express and develop probabilistic ideas; and\ud m to describe and analyse how the tool-based game mediated the children's\ud expressions of chance events.\ud An open computer game was designed for children to express understandings of\ud randomnessa s formal conjectures, so that they were able to examine the consequenceso f\ud their understandings. The game was designed simultaneously to afford children the\ud opportunity to explore and express their intuitions and ideas, and to give the researcher the\ud opportunity to study how probabilistic ideas evolved during the activity.\ud The study was organised in two main phases. The first, iterative design phase, compared\ud two cycles of design and experiments with children as they played with and reconstructed\ud the game. The second phase consisted of the learning investigation phase, which describes\ud in detail the expressed ideas of children in using the game. This thesis shows how a visible\ud and 'continuous' medium, i. e. one in which the sample space is represented by a spatial\ud and dynamic metaphor, can enhance young children's expressions of randomness. The\ud findings identify children's initial meanings for expressing stochastic phenomena and\ud describe how the computer tool-based game helped to shift children's attempts to\ud understand randomness from looking for ways to control random behaviour, towards\ud looking for ways to control events. This was significant, since the study analyses howchildren constructed their own ideas for fairness and in particular, how they constructed\ud both symmetrical and asymmetrical spatial arrangements for it. In general, it is conjectured\ud that the structure of the game, and in particular, the linkage between its components,\ud assisted children in developing associated mental structures that developed their\ud understandings of chance. Finally, evidence is presented that the children constructed a set\ud of 'situated abstractions' for ideas such as 'distribution' and the 'law of large numbers'