Preschool children's problem solving processes are investigated in both direct and indirect ways. Direct investigations focus on substantive and methodological issues related to how children solve a few well defined puzzles, such as the Tower of Hanoi and the Tangram. Indirect investigations deal with related issues: U-shaped (or non-monotone) developmental curves, rates of processing, structure-process invariance, and instructional theory. Findings indicate that by the time children reach kindergarten, they appear to have acquired without direct instruction variations on many of the components of mature problem solving strategies. Therefore, attempts to instruct children to be better problem solvers must first make a careful determination not only of the level of their performances, but also of the strategies they use. A methodology involving the characterization of children's knowledge in terms of rules has been developed to facilitate such a determination. The position is taken that U-shaped curves always reflect an artifact of the assessment procedure, do not reflect any interesting underlying processes, and ultimately must be accounted for by general mechanisms of self-modification that are neither constrained nor informed by U-shaped phenomena. The focus on rates, processes, and structures as potential sources of developmental differences maps the domain for further investigations of how children learn to solve problems. (Author/RH)