Sophian, C., Garyantes, D., & Chang, C. (1997). When three is less than two: Early developments in children's understanding of fractional quantities. Developmental Psychology, 33, 731-744.

Four experiments examined young children's understanding of the inverse relation between the number of parts into which a quantity is to be divided and the size of each part. In Experiment 1, 5-, 6-, and 7-year-old children tended to judge, incorrectly, that bigger shares would result from sharing with more, rather than fewer, recipients. In Experiment 2, 5-year-olds correctly recognized the inverse effect of additional recipients when the sharing was based on subtraction rather than equal partitioning. In Experiment 3, a modification of the equal-sharing task from Experiment 1 designed to reduce cognitive complexity was successful in eliciting correct performance from 7-year-olds but not from 5-year-olds. However, 5-year-olds showed marked improvements when they were given a chance to compare the outcomes of sharing with different numbers of recipients. Experiment 4 corroborated and extended this evidence of learning.

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