From among all of the problem solving strategies that children can use, how do they decide which ones to use? Illustratively, when children spell a word, sometimes they retrieve a spelling and write it, other times they sound out a spelling phonetically still other times they look up the word in a dictionary. Even preschoolers appear quite adept at making adaptive strategy choices, that is, at making choices that meet the twin goals of accuracy and efficiency. The goals of the present research project are to specify how children decide which strategy to use, how the ability to make such choices develops, and what functions the strategy choice process serves. A variety of methodologies will be used to pursue these goals. The basic one will involve making videocassettes of children's performance, and examining their audible and visible behaviors between presentation of the problem and statement of the answer. The videocassettes will help in identifying what strategies children are using, how long each strategy takes to execute, and how accurate it is. Supplementing these empirical techniques will be two types of formal modeling: mathematical modeling of the strategy choice process, and computer simulation of the way in which the process develops. The proposal involves the extension of work on children's strategy choices in five domains: addition, subtraction, multiplication, time telling, and reading. Among the issues that are to be explored are the relations between rule-governed and associative knowledge, how children integrate new strategies with existing ones, the effects of use of backup strategies on the development of efficient retrieval, and the existence of unities across domains in the empirical relations that emerge from children's strategy choices, the mechanisms that underlie performance and development in each domain, and the sources of individual differences. Several quite applied problems also will be examined, such as whether it is a mistake to forbid children to use their fingers to add and subtract.