A key question in early number development is how 4- and 5-year-olds learn the roles that counting and cardinal numbers play when comparing quantities. Children who wrongly used length to identify numerosity were assigned to five experimental groups and trained to judge whether a puppet - who sometimes miscounted - created equivalent sets. Over three training sessions, children who were asked to compare sets after they were counted learned to base their judgments on cardinal numbers when the puppet counted accurately by being given feedback. However, only the groups who were also asked to explain either their own or the experimenter's reasoning made progress in identifying the puppet's miscounts. This ability to recognize the importance of counting accuracy for quantitative comparisons predicted whether children would spontaneously count to compare sets on a post-test. The importance of asking children to identify miscounts is discussed alongside the social factors that influence children's recognition of the relationship between procedural counting, cardinality and relative number. © 2007 The Authors. Journal compilation © 2007 Blackwell Publishing Ltd.