A framework is presented for characterizing competence for cognitive tasks, with a detailed hypothesis about competence for counting by typical 5-year-old children. It is proposed that competence has three main components that are called conceptual, procedural, and utilizational competence. Conceptual competence, which is discussed in greatest detail, is the implicit understanding of general principles of the domain. Procedural competence is understanding of general principles of action and takes the form of planning heuristics. Utilizational competence is understanding of relations between features of a task setting and requirements of performance. A characterization of conceptual competence for counting is presented, in the form of action schemata that constitute understanding of counting principles such as cardinality, one-to-one correspondence, and order. This hypothesis about competence is connected explicitly to a detailed analysis of performance in counting tasks. The connection is provided by derivations of planning nets for procedures that are included in process models that simulate children's performance. It is concluded that the nature of young children's understanding reflects competence that supports the understanding of counting as well as later development, such as explicit understanding of the role of one-to-one correspondence in definitions of equivalence. (Author/JN)