This paper reports a study to address two questions concerning children's understanding of average: How do children construct and interpret representativeness within the context of data sets? and How do children think about the mean as a particular mathematical definition and relationship? Twenty-one students (seven each of 4th, 6th, and 8th graders) were interviewed using a series of seven open-ended problems that examined the notion of average. The four that yielded the most results were identified and included in the report. These were two "Construction Problems," an "Interpretation Problem," and a "Weighted Means Problem." Analysis of transcripts and summaries of the interviews produced five approaches that children used for constructing and describing average. The approaches were: (1) Average as Mode; (2) Average as Algorithm; (3) Average as Reasonable; (4) Average as Midpoint; and (5) Average as Mathematical Point of Balance. Interpretation of these approaches concluded that students whose strategies were dominated by Modal or Algorithmic approaches did not view average as a representative measure of the body of data as a whole. Interpretation also concluded that students whose strategies were dominated by the Reasonable, Midpoint, and Balance approaches embodied different aspects of constructing a mathematical definition of average. Further research questions resulting from the research are discussed. (Contains 11 references.) (MDH)