This report describes the first year of an ongoing intervention study and the learning of a group of teachers who are working together to make changes in the ways they teach mathematics. The document presents a vision of mathematics education based on research in cognitive psychology which indicates that learners construct knowledge rather than being passive receivers of knowledge. 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) 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 non-traditional approaches in the teaching and learning of mathematics. (LL)