Children learn arithmetic procedures by rote, rather than by constructing them with an understanding of numbers. Rote learning produces lack of flexibility, nonsensical errors, and other difficulties. Proposed is a theory of conceptual understanding and its role in learning and executing arithmetic procedures. The basic hypothesis is that principles constrain the possible states of affairs, thereby enabling learners to monitor their own performance and correct errors. A new knowledge representation is proposed, the state constraint. The theory has been implemented in the Heuristic Searcher, a computer model that learns arithmetic procedures on the basis of general principles encoded as constraints on search states. Simulated is: (1) the discovery of a general counting procedure in the absence of either instruction or solved examples; (2) flexible adaptation of a counting procedure in respose to changed task demands; and (3) correction of subtraction errors in the absence of external feedback. The theory provides novel answers to several questions on conceptual understanding, generates testable predictions about human behavior, deals successfully with technical issues, and fares well on evaluation criteria. Future work will focus on how knowledge and experience interact in procedural learning. Over 110 references are included. (Author/MNS)