Considerable discrepancies between the mean and median often occur in data sets that are skewed left, skewed right, or have other unusual features. In such cases, it is important to analyze the data and context carefully to decide how best to describe centers of distributions. The importance of this type of statistical thinking is acknowledged in the grade 6 Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010): Students are to engage in "relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered" (CCSSI 2010, p. 45). Given the steep challenge set in the grade 6 CCSSM, the authors set out to investigate how students' reasoning about choosing appropriate measures of center might be developed. They conducted a classroom research study during a summer mathematics program funded by the National Science Foundation. The study focused on a small group of students: two females and two males. It involved intensive data gathering and analysis using videorecorded clinical interviews over a period of nine weeks. In this article, the authors share the lessons they learned about developing students' abilities to choose appropriate measures of center.