The study discussed here aims to describe students' understandings of the definition of a mathematical function, which was achieved through a pilot case study of clinical interviews with four participants -- two ninth graders and two twelfth graders. The participants were recruited from the same urban public high school in the northeast of the United States, which serves a diverse racial, ethnic, and cultural community. All four participants were selected by their mathematics teachers because of their high grades and skill level. The participants were interviewed individually about questions pertaining to the definition of a mathematical function. Analyses of the interview responses revealed that the twelfth graders understand the definition of a mathematical function differently than do ninth graders. [For the complete proceedings, see ED658295.]