In Experiment 4, children were tested again with large

numbers, but with transformations that did not affect one-to-one correspondence mappings, therefore removing the burden of having to update this mapping. As in Experiment 2, the transformations involved removing or adding one puppet to a box containing either 5 or 6 puppets. Two types of events were presented to the children. In the identity condition, one puppet first exited the box and then returned to the box after a short delay. At the end of the trial, the final set beta-catenin assay was thus composed of exactly the same individuals as at the start of the trial. The substitution condition differed in that the puppet returning to the box was a different individual from the puppet that left the box: This event thus preserved the number of elements in the set but not the identity of all its individual members. If children were not able to combine information about one-to-one mappings with information about transformation events, for example by failing to remember both pieces of information at the same time, then they should fail to distinguish between the events involving 5 vs. 6 objects in either condition. If children interpreted one-to-one correspondence as establishing numerical equivalence (i.e., if they realized that additions and subtractions affect one-to-one mappings and that substitutions

do not) but failed to compute the updated one-to-one mapping in the addition/subtraction conditions of Experiment 2, then they should succeed in both the Screening Library identity and the substitution conditions. Finally, if children could use one-to-one mappings to establish a correspondence relation among specific objects, but not to establish numerical equivalence, then they should succeed in the identity condition but fail in the substitution condition. Participants were 24 subset-knowers (16 female, mean age 34.04, 32:11–35:22). Displays were the same as in Experiments 1 and 2. A 6-branch tree was used, with sets of 5 or 6 puppets. Children received 4 experimental trials: two trials with sets of 5 puppets, and two trials with sets of 6 puppets, presented in a semi-alternating order as in past experiments.

In both the identity and substitution conditions, the transformation event started with a puppet taken out of over the box. In the identity condition, this puppet was returned to the box after narrating a cover story (“He is going to get a snack”). The cover story varied between the first and second pair of trials in an effort to maintain interest. The events in the substitution condition resulted in the substitution of one puppet by another identical puppet. Again, two story lines were used for the first and second trial pairs. In the first pair of trials, the substitution was enacted as a subtraction followed by an addition. The experimenter first took a puppet out of the box and placed it in a bag on the floor, narrating, “He does not want to sleep; he is going to the jungle”.