In the research reported in this paper, we address two major sets of questions about children's understanding of average. 1) When they are working with data sets, how do children construct and interpret indicators of center? It's important to examine how children learn to describe data sets in a meaningful, useful, and flexible manner. In particular, we are concerned with the development and use of the idea of "representativeness" in the context of real data sets. The second major question we are addressing deals with the use of the mean in a precise mathematical sense: 2) How do children develop their thinking about the mean as a mathematical relationship? This question moves into the important more general question of how children develop mathematical abstractions, and how they map (or fail to map) these abstractions onto their informal understanding of a concept.