The problems used in the study involve two collections of elements\ud of two colours. The proportions of elements of each colour in each of\ud the collections is varied, and the way children reason when asked which\ud collection they would prefer in order to gamble for a specified outcome\ud is investigated in three situations:\ud (a) The elements are beads to be drawn from boxes. (72 subjects\ud aged 5-10 years, 48 subjects aged 11-14 years).\ud (b) The elements are single segments marked on circles of different\ud sizes with pointers to be spun. (72 subjects aged 6-11 years).\ud (c) The elements are similar to (b), but marked into separate\ud pieces to allow comparison by counting. (60 subjects, aged\ud 6-10 years).\ud Four possible ways of solving such problems are outlined:\ud Method 1: Guessing, alternating choices and other irrelevant methods.\ud Method 2: Comparing the amounts of the target elements in each collection,\ud and choosing the collection with the greater amount.\ud Method 3: Comparing the differences between the amount of target and non-target\ud elements in each collection, and choosing the collection\ud with the most favourable difference.\ud Method 4: Comparing the proportions of target and non-target elements\ud in each collection, and choosing the collection with the most\ud favourable proportion.\ud Within the main age range investigated. (6-10 years), methods 1-3\ud are found to form a developmental sequence, in situation (a), whereas in\ud situations (b) and (c) the predominant developmental sequence is from\ud Method 1 to Method 2 only. It is argued that this can be explained by\ud considering the methods of quantification used by subjects in each\ud situation.\ud (A summary of the way in which the main themes are developed in the\ud thesis is given at the end of the thesis.