Early results are reported for an effort to use a reasoning-based mathematics program in the primary classroom. In the preschool years, children develop knowledge about quantities of material, referred to as protoquantitative reasoning. Integrating counting with the protoquantitative schemas seems to occur as early as 4 years of age. Drawing on this knowledge of children's abilities, the following set of principles was developed for the primary arithmetic program: (1) develop children's trust in their own knowledge; (2) draw children's informal knowledge into the classroom; (3) use formal notation as a record of discussions and conclusions; (4) introduce key mathematical structures as quickly as possible; (5) encourage everyday problem finding/solving; and (6) talk about mathematics, rather than just doing arithmetic. These principles have been applied to two cohorts of children, one beginning in the first grade, and the other beginning in the second grade. In these two classes, the program appeared effective in raising mathematics scores on the California Achievement Test for children of all ability levels, and the approach was successful for educationally disadvantaged children. The program is described as a version of the cognitive apprenticeship called for by A. Collins and others in an influential paper (1989). These initial efforts will be part of a long-term project to revise instructional practice. Seven figures and a 12-item list of references are included. The paper's discussant is Nancy Vye (and others) in a taining piece entitled "Exploring a Reasoning-Based Program for Teaching Arithmetic". (SLD)