This study was undertaken to gain insights into children's understanding of rational numbers as quantities; that is, the extent to which they associate a size with a fraction like 2/3. Eight children in an experimental group in DeKalb, Illinois, chosen to reflect the range from low to high ability, were observed during 30 weeks of experimental instruction during grades 4 and 5. A classroom-sized group of 34 middle-ability children in grades 4 and 5 in Minneapolis simultaneously took part in the same teaching experiment, providing children with manipulative-oriented instruction. Seven interview assessments, each preceded by about 4 weeks of instruction, were videotaped with each DeKalb child and with eight Minneapolis children. Written tests were also given. Data from the three fifth-grade assessments are included in this report. The three tasks are described, and children's reactions are reported in detail. They had varying success on the tasks. It appeared that three knowledge structures are essential for the development of a quantitative understanding of rational number: estimation, fraction equivalence, and rational-number order. These structures appeared to develop somewhat independently, but need to be coordinated for success with rational number situations. (MNS)