This project was a one-year study of California's reform effort in mathematics education. The research focused on two issues of great importance to current efforts to reform mathematics instruction: (1) What distinguishes teaching mathematics for "understanding" from rote performance and what does teaching mathematics for understanding in elementary schools entail? and (2) What can policy do to improve such mathematics teaching and learning? The research addressed these questions by examining effects on classroom practice of a state policy to promote teaching mathematics for understanding. In 1988-89 researchers observed and interviewed 23 second- and fifth-grade teachers concerning their knowledge and beliefs about mathematics teaching and policy initiatives, their efforts to change their practice, and resources and constraints that influenced their ability to do so. Researchers also interviewed administrators and policymakers at the school, district, and state levels. This report focuses on changes in teaching practices related to the state policy of teaching mathematics for understanding. Detailed case analyses are presented of nine elementary teachers in six different elementary schools (three high SES and three low SES schools) in three different school districts in California. Three themes emerging from these cases are discussed in a final commentary. The first theme concerns texts and other curriculum materials as agents of change; the second theme explores the "layers of reform"; the third theme focuses on the profound influence of practice on policy. (KR)