This is one of a series of units intended for both preservice and inservice elementary school teachers to satisfy a need for materials on "new mathematics" programs which (1) are readable on a self basis or with minimal instruction, (2) show the pedagogical objectives and uses of such mathematical structural ideas as the field axioms, sets, and logic, and (3) relate mathematics to the "real world," its applications, and other areas of curriculum. This unit explores basic counting and order concepts primarily from an informal, but essentially logical point of view. Part 1 describes three experiments performed by Piaget et al., in order to understand better the process a child goes through in attaining a working familiarity with number concepts - (1) an experiment involving the basic conception of "more than" without number ideas, (2) an experiment involving one-one correspondence, and (3) an experiment involving concepts on sets and cardinal numbers. Part 3 shows the inter-relationship between counting and order. (RP)