Discusses the first session and events from other sessions of a mathematics circle consisting of the author and four preschool children. Presents discussions that ensued with the children when asked to solve problems related to Piaget's stages of cognitive development. (MDH)

Knowledge about the structure of arithmetic expressions enables people to reason effectively about such expressions, including an ability to judge equivalence under transformations. This paper reports an empirical study of six middle-school children who judged the equivalence of three sets of three-term arithmetic expressions with an addition and…