Mathematics learning consists of more than simply memorizing rules and procedures. Unfortunately, much of the mathematics currently being learned is of this type. A concensus is emerging, however, that a necessary condition for students to understand important mathematical concepts is that they posess multiple representational systems for those concepts and be able to translate smoothly and efficiently among them. This project will study how we can teach effectively for students' development of multiple representational systems and how such instruction affects their development of concepts and skills. Two computer programs will be designed for this project; one for decimal numeration and one for algebra. Both programs will present simultaneous, "linked" representational systems. Students will solve problems by acting directly upon one representational system with their actions being automatically reflected in the other. Operations on decimals numbers will be studied for five weeks with a class of seventh graders. Concepts of expression, identity, linear equation, and solving linear equations will be studied with pre-algebra eighth graders over a school year. The basic research methodology will be that of the constructivist teaching experiment. Through carefully planned individual and small group interview protocols, a model (models) of student knowledge will be constructed which describes the relationship beween the representations taught. Curriculum materials to facilitate mathematical understanding will also be developed. The proposed research is challenging and should lead to significant advances in mathematics and science education. The Principal Investigator is highly regarded in the mathematics education community and this is reflected in this well conceived research project.