Considered is a conceptual analog of buggy algorithms and rule-based mathematical development. The investigations consider whether children's efforts to make conceptual sense of new mathematics instruction in terms of their available knowledge may sometimes lead them to make systematic errors. In particular, the possibility is explored that children overgeneralize concepts from a familiar domain of mathematics in order to interpret a new domain. A total of 113 children in the early phases of school instruction on decimal fractions participated: an American sample from grade 5 (n=17); an Israeli sample from grade 6 (n=21); and French samples from grades 4 (n=37) and 5 (n=38). Children were individually interviewed using described tasks. The findings are discussed in some detail. There seem to be fundamental differences in the kinds of conceptual understanding that produce the Whole Number rule and Fraction rule. Different patterns of rule categorization among the three countries indicate that different curriculum sequences produce different patterns of rule invention. Errorful rules appear intrinsic to learning and cannot be avoided in instruction: they can best be regarded as useful diagnostic tools for instructors, to detect the nature of children's understanding of a mathematics topic. (MNS)