Research recurrently indicates that children who have difficulty with arithmetic often use systematic routines that yield wrong answers. Recent research has focused less on identifying the most common errors among groups of children and more on analyzing individual children's errors. This paper considers the source of systematic errors in subtraction with multidigit numbers and appropriate instructional responses. The author argues that conceptual misunderstandings lie at the heart of many errorful procedures, and that these misunderstandings are what should be addressed in instruction, as well as the problem of linking conceptual understanding to procedural skill. Following an analysis of the nature of "buggy algorithms" in subtraction, teaching the semantics of procedures is considered. Principles of subtraction and place value are presented, and children's knowledge of the principles is reviewed. Instructional experiments on how the principles apply to written arithmetic are described. Finally, conclusions and some questions are presented. The importance of error analysis research is confirmed. Systematic errors probably arise from a basic failure to mentally represent arithmetic procedures in terms of operations on quantities within a principled number system, rather than as operations on symbols that obey largely syntactic rules. (MNS)