This is an empirical inquiry concerning children's concept development and early mathematics teaching. The intention is to broaden the understanding of preschool children's perceptions of the concept "half" (as 1 of 2 equal parts of a whole), in designed mathematics teaching settings. Three teachers working with 4-5-year-old children participate in an in-service program, involving continuous and cooperative reflection and theoretically designed teaching activities. Observations of pedagogical strategies and children's responses reveal that: children show qualitatively different ways of experiencing the same concept; the ways of experiencing are critical for how the intended learning object is perceived; and the dimensions of the learning object are related to each other, suggesting a hierarchical organization of how to perceive aspects of "half."