In this study, Piagetian theory is extended to explore ways in which children construct their understanding of our notational system and place value. Eighty children aged 4 to 9 were asked to group objects, draw pictures of grouped quantities, write numerals to indicate amounts, and theorize about the relationship between their written numerals and drawn quantities. Developmental levels were inferred for these and other tasks, and level x age analyses were performed. Eighteen hypotheses that children brought to bear on the meaning of the notational marks in relation to the symbolized quantities were formed, and were grouped into five developmental levels. Among the patterns that emerged was a developmental sequence in the kinds of ideas used by children, singly and in combination. Children's understanding of place value seems to be built in phases over a long period of time, in conjunction with other kinds of knowledge. Some developmental relations were evident; however, the children appeared to have many theories that intruded upon their understanding of the numeration system. Some parallel findings between the historical development of our system and the developing knowledge of that system in children are pointed out. A selection of the children's graphic productions is appended. (Author/MNS)