This study investigated the relationships between Piagetian logical reasoning abilities with an information processing capacity, and first-grade children's performance on verbal addition and subtraction problems. An analysis of simple arithmetic problems indicated that several reasoning abilities identified by Piaget are needed to achieve operational understanding of addition and subtraction. The abilities studied were number conservation, class inclusion, and transitive inference. The arithmetic problems used were varied systematically in semantic structure, number size, and the availability of objects to aid the solution process. Statistically significant differences between the developmental groups were found for some of the cognitive variables on some problem types. However, no clear pattern emerged to suggest that a particular cognitive ability was especially important for solving a specific problem type or using a specific solution strategy. None of the cognitive abilities was absolutely required to solve any of the problem types or to use a given strategy. The report indicates that the use of these studied cognitive tasks as readiness variables for arithmetic instruction should be questioned. The study also suggests that mathematics problem solving ability may need to be reassessed for each instructional topic. (MP)