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This study examines children's efforts to make sense of new mathematics instruction. The study documents major categories of errors that appear consistently as children learn decimal fractions. It then establishes the conceptual sources of these errors. Whole number errors derive from children's applying rules for interpreting multidigit integers. Fraction errors derive from children's efforts to interpret decimals as fractions. Different curriculum sequences influence the probability that these classes of errors will appear. It is suggested that errors are a natural concomitant of students' attempts to integrate new material that they are taught with already established knowledge. Since errorful rules cannot be avoided in instruction, educators are encouraged to use them as useful diagnostic tools to detect the nature of children's understanding of a mathematics topic.