Presents results of a project with second and third graders to establish concepts of equality that led to the realization that teachers often help students too much and may push them algorithmically beyond their ability without the development of proper referents. (Author/MKR)

Describes the components of, and theoretical/research bases for, learning cycle pedagogy, which is especially appropriate for the special needs of the middle-school student. Three stages of the learning cycle are exploration, conceptual invention, and expansion. (Author/MKR)

Reviews research using diagrams, number sentences, and algorithms to help students learn to add and subtract; poses questions on the relationship of instruction to children's knowledge construction; and proposes a research agenda in this area. (86 references) (MKR)

Presents an episode in which a conflict arose as sixth-grade students attempted to create nonstandard units of measurement. Discusses the conflicting requirements of language and symbol use in mathematical communication in the activity. (MKR)

Illustrates the multiple strategies used by second graders to solve a problem with three addends and how their teacher tries to map their thinking into the system of mathematical notation. Describes the American Federation of Teachers' Thinking Mathematics program that the teacher uses. (MKR)