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In this paper, we attempt to clarify what it means to teach mathematics for understanding and to learn mathematics with understanding. To this end, we present an interactional analysis of transcribed video recordings of two lessons that occurred in different elementary school classrooms. The lessons, which are representative of a much larger data corpus, were selected because both focus on place value numeration and involve the use of similar manipulative materials. The analysis draws on Much and Shewder’s (1978) identification of five qualitatively distinct types of classroom norms and pays particular attention to the mathematical explanations and justifications that occurred during the lessons. In one classroom, the teacher and students appeared consistently to constitute mathematics as the activity of following procedural instructions in the course of their moment by moment interactions. The analysis of the other classroom indicated that the teacher and students constituted mathematical truths as they coconstructed a mathematical reality populated by experientially real, manipulable yet abstract mathematical objects. These and other differences between mathematical activity in the two classrooms characterize two distinct classroom mathematics traditions, one in which mathematics was learned with what is typically called understanding and the other in which it was not.