Preschool children are often good at counting things but seem slow to learn that there is more to counting than simply finding out how many are in a single set. Counting is useful when comparing sets and when creating new sets to match existing ones. This is part of the numerical understanding that educators wish to foster in schools. In two experiments we found that although counting proficiency was necessary to solve these types of arithmetical problems, the best predictor of success was the ability to detect and reason about someone else's miscounts. We argue that these data suggest that a breakthrough in connecting counting with arithmetical problem solving is associated with an explicit representation of the rules governing cardinality and how they are applied correctly or incorrectly. We propose that teaching 4-year-olds to compare sets should include tasks where they judge another's counting proficiency, giving them the opportunity to conceptualise miscounts.