Most experts believe that young children possess a substantial amount of informal knowledge about mathematics. The teacher's role is to create a link between children's ability to use informal math and the ability to understand the more formal math taught in elementary school. Teachers must help children construct and elaborate upon what they already know, so they can "re-invent" mathematics for themselves. A reflective teacher helps the child discover and communicate ideas that would not have occurred spontaneously without the adult's help. This paper describes mathematics activities in two early childhood classrooms--a preschool class and a kindergarten class. Activities in the preschool class include children's books on counting, simple and repetitive songs, counting and spatial games, and learning centers such as the matching center and block center. The Kindergarten class uses a circus theme to cover topics such as number and measurement, sequence and ordering (time), measurement/weight, and part-part-whole. The paper then uses these descriptions to discuss important aspects of early childhood mathematics instruction, including the teacher's role and the development of spatial and number sense. Contains 13 references. (EV)