Many preservice teachers come into teacher education programs with a rule-based view of mathematics, a view that is to a large extent indicative of their previous school experiences. In teaching preservice elementary school teachers, this author has used his instrumental understanding as the starting point for rich discussions about the fundamental ideas of mathematics and meaningful mathematics teaching. In this article, he focuses on preservice elementary school teachers' reconstruction of meaning for the midpoint formula, highlighting the interplay between instrumental and relational understanding (Skemp 1978)--in other words, the interplay between procedural and conceptual knowledge (Silver 1986) in learning mathematics for teaching. The author further provides alternative perspectives on the relationship between midpoint and average to suit the needs of readers at various levels of mathematics learning and instruction.