In this article I detail the conceptual trajectory of a classroom of 2nd- and 3rd-grade students as they reinvent topographical lines to represent height in a map within the constraints of an overhead perspective. In my analysis I pay special attention to the role of social interaction--and in particular the role of the teacher--in the process of knowledge production. First, I demonstrate how the invention of representational forms by individuals occur as part of a larger social process of creating cultural conventions and negotiating a taken-as-shared understanding of these new tools. Second, I show how gesture, as a part of the larger semiotic ecology for meaning making around representations, contributes to creation of understanding. Third, I make some preliminary proposals regarding the process of transforming personal inventions into cultural conventions. The analyses are intended to contribute to our field's growing understanding of young children's activity when inventing representations (i.e., metarepresentational competence), the mechanisms for learning within instructional activities based on the iterative refinement of these representations (i.e., progressive symbolization), and a rejection of the dichotomy between an individual's cognition and her participation within a cultural community.