This thesis addresses the difficulty young children have making the transition from solving simple addition and subtraction problems with referential support to the formalised language of arithmetic ('what does one and one make?') (Hughes, 1981, 1986). Nine experiments are reported testing a total of 782 children between 4 and 5 years old on their use of strategic prompts in addition and subtraction. Results produced evidence for their significant positive impact, whether it be imagery, solving problems with their eyes closed or provision of a memory-check for the initial addend. Effectiveness was mainly on more difficult problems, particularly subtraction, with one mentalistic strategy generating results as effective as those using concrete referents. Five-and-a-half year old children were shown to have greater sensitivity overall to such prompts, whereas the younger children of around five years showed less sensitivity to the imagery-prompt, but like their older counterparts were most sensitive to prompts involving a memory-check. Referential support was also found to be effective when the older children were presented with written formal code notation. Furthermore, children were also found to be sensitive to implicit (rather than explicit) strategic prompts with child-initiated strategies, like 'fingers', most effective when chosen by the children themselves. Results are discussed in relation to how such explicit strategic prompts are effective, whether working directly or indirectly.