This study explored fourth-grade teachers' understanding of mathematical problem solving, an aspect of the Vermont portfolio assessment program that has been largely ignored. Teachers' conceptions of problem solving, their knowledge of problem-solving strategies, their selection and evaluation of problem-solving tasks, and their instructional practices related to problem solving were examined in a representative sample of 20 fourth-grade teachers. Data were collected through a written survey and a structured telephone interview. Results indicated that Vermont teachers believe that the program has taught them much about problem solving and the everyday applications of mathematics. They have learned much of what has been communicated through the state training materials and network meetings. However, they do not share a common understanding of problem solving and do not agree about which skills are most essential. Teachers tend to rely on the state scoring rubrics for practical guidance. Differences in understanding lead to differences in practice, which should be addressed through continued professional development. Three appendixes present the scoring rubric, the survey and interview protocol, and a description of characteristics of good problems. (Contains 3 tables, 8 figures, and 21 references.) (SLD)