We have studied first- and second- grade mathematics classes in Japan and the United States in which story problems and problems about mathematical concepts were used to promote higher-level thinking. Our re earch leads us to question the common a umption that teachers should aim each le son at a particular conceptual goal for all pupils and expect nearly all of them to succeed and to understand the same concept. Instead, a teacher's aim should be as close as possible to accepting conceptual diversity or “splatter” among pupils. We believe that steady work on such an aim through problem solving and dialogue will lead to outstanding conceptual power in pupils. In this article, we present three examples from lessons that reflect this aim, as well as suggestions that we have drawn from these and other lessons.