This report describes the first year of an ongoing intervention study and the learning of a group of practicing teachers working together with researchers to make changes in the ways they teach mathematics. When reflecting on their beginning efforts, the researchers found that several assumptions in their planned interventions were not borne out by the group's experience. The subject matter content (integers) they selected as an initial focus of study posed major difficulties for several of the teachers. The teachers' interest in talking about their own practices strongly influenced the group's interactions. Collectively, the teachers and researchers created a learning community that was grounded in watching videotapes of mathematics teaching in a third grade classroom and discussing ideas about teaching and learning in ways that were different from their own experiences as teachers and students. Here the researchers pose a set of conjectures that serve as a framework for the continuing collaboration of this group of educators' inquiry into nontraditional approaches in the teaching and learning of mathematics. (Author)