This paper deals with one aspect of the endeavor to generate a theory of the development of mathematical thinking of children in the early years ages 3 to 10. By comparing two scenes, one from preschool and one from a first grade mathematics class, the relationship between diagrammatic and narrative argumentations among children and teachers is reconstructed and related to possible developmental trajectories of mathematical thinking. Theoretically, I attempt to implement these developmental paths in a concept of an "Interactional Niche in the Development of Mathematical Thinking." (author abstract)