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The current research aimed to examine children’s understanding of cardinality by looking at their ability to use several quantitative concepts that underpin this understanding: correspondence, counting and equivalence in the context of sharing. Understanding cardinality requires children to develop knowledge about the relations between these quantitative concepts which is important for the development of mathematical reasoning. The first study aimed to investigate how flexibly children can use correspondence to build equivalent sets in different types of sharing scenarios: equal sharing, reciprocity and equity. In some situations two characters each received one object at a time, and in others one character received double units while the other character received single units. After children shared blocks between the two characters, they were asked to make a number inference about the cardinal of one set after counting a second, equivalent set. Children had more difficulty sharing in the reciprocity and equity conditions than the equal sharing condition. The majority of children were able to make number inferences in the equal sharing and reciprocity conditions where both characters received equivalent shares in the end. A second study with new groups of four and five- year-olds investigated whether children were using visual cues about the relation between double and single blocks to help build equivalent sets and make number inferences. It was predicted that the use of coins would be difficult and would increase the difference between the equal sharing and reciprocity conditions. In half of the trials children shared Canadian $2 and $1 coins and in half they shared blocks. There are no visual cues about the relation between $2 and $1 coins because they are the same size. Children were allowed to use counting or correspondence to build equivalent sets to compare their use of both strategies. Contrary to the first study, the reciprocity and equal sharing conditions were not significantly different. This may be due to the appearance of a new sharing strategy in the reciprocity condition termed “equalizing” where children first counted each set, dealt singles to make the two sets equal and then shared blocks or coins on a one-to-one basis. There was also no significant difference between the trials using coins and trials using blocks. The majority of children were able to answer the number inference questions correctly, however 25% of children made the number inference after sharing all singles but not after sharing doubles and singles, suggesting that using different units did impact their understanding of the equivalence of the two sets. A third study aimed to investigate children’s ability to coordinate cardinal and ordinal information to determine the cardinal of a single set, and their ability to coordinate counting principles with knowledge of equivalence to determine the cardinal of an equivalent set. Children in this study were asked to make a numerical inference about a set of blocks after watching a puppet correctly or incorrectly count an equivalent set of blocks. Many children were able to identify that the puppet did not count correctly, but struggled to correct the mistake. This indicates a gap in their knowledge about ordinality and cardinality in the context of a single set. The miscount also impacted their ability to make a correct number inference. Children performed significantly better on trials where the puppet counted correctly than trials where he made a counting error. This suggests that while children have good knowledge of counting principles in isolation, they are still developing an understanding of how to coordinate these principles with ordinal information and knowledge of equivalence to establish the cardinal of one set and to infer the cardinal of an equivalent set.