Although the average, or arithmetic mean, has a rich conceptual meaning, it is often defined simply as the outcome of a procedure. The purpose of this study was to compare the nature and extent of the procedural and conceptual understandings developed by two groups of students who received different forms of instruction, one based on the traditional numerical algorithm, the other on a visual algorithm. When confronted with tasks varying along several dimensions, students adjusted or extended their basic approach to finding the arithmetic mean in ways that give insight into their understanding of the concept. While both groups of students showed a degree of understanding and flexibility with the procedure they had been taught, students who had learned the visual procedure showed a deeper conceptual understanding of the arithmetic mean. An appendix contains selected tasks presented to subjects. (Author/MKR)