Findings from prior research are drawn together to create a learning model for elementary school mathematics in the cognitive-constructivist tradition. A potential teaching/learning process consistent with the model was developed and applied in a longitudinal collaborative arrangement between university personnel and a local elementary school using a conceptually based curriculum that posed problems requiring active student involvement with physical materials to model mathematical situations, defined symbols, and developed solution strategies. As children used these materials, they actively construed the operations and principles of arithmetic. In another phase children sketched the materials and situations in a move toward abstraction. They then constructed mental images through imagining actions on physical materials. Experiences with the mental images allowed for student construction of arithmetic generalizations and problem solving skills. The computer served as another tool for constructing methods of dealing with problems. The conceptual frame of cognitive constructivism appears to provide for an awareness that understanding in elementary mathematics must involve the active search for, creation of, and use of links between the abstractions and generalizations and the world of personal experiences. Eleven figures illustrate the model and the discussion. There is a 38-item list of references. (SLD)