This research paper proposes several mathematical models which help clarify Piaget's theory of cognition on the concrete and formal operational stages. Some modified lattice models were used for the concrete stage and a combined Boolean Algebra and group theory model was used for the formal stage. The researcher used experiments cited in the literature and demonstrations to determine the appropriateness of the models. With some reservations which are sited in the research, the mathematical models were quite consonant with the cognitive stages they were designed to describe. (Author)