This report describes an investigation of how young children respond to two types of tasks: (1) finding one-half of a continuous and a discrete material; and (2) attempting to share continuous and discrete material equally between two dolls. Continuous material, such as string, paper, or liquid, is quantified by adults using measurement units. A question often asked about this kind of material is "How much?" Discrete material, such as buttons or beads, is quantified by counting--asking "How many?" The investigation was interested in both the accuracy of the children's responses and the methods they used. The performance of one child, Julie, classified as a weak "halver," was examined in detail to explore how social practices are bound up with rational number knowledge, including serial sharing and parallel sharing. Based on findings, the report asserts that if people want young children to learn mathematics in a meaningful way, it is important to identify situations in which quantities can be shared using methods familiar to children. Sources of children's knowledge, experience, and motivation to subdivide quantities include the primitive personality of the child, the modeling behavior of parents and other adults, and interactions with siblings and peers. The report further explores children's abilities in subdividing quantities by discussing distributive counting, the process by which children systematically allocate items resulting in equal shares. The report discusses the different ways in which children "deal," or the strategies they use to equalize shares, then describes the unique role the fraction one-half plays in the development of children's rational number knowledge. A second study examining strategies used by preschoolers when counting and sharing is also noted. The report concludes that teachers should be able to involve young children in problems of sharing even if those children have not yet become rational counters, and that sharing tasks may assist children's developing counting skills through opportunities to check or verify the sizes of shares. (Contains 34 references.) (EV)