Results are now available from the Third International Mathematics and Science Study (TIMSS) with its 5 main components, 41 cooperating countries, over 500,000 participants, and coverage of the full spectrum of mathematics and science from grades 4 through 12. American educators, parents, and policy makers have found the results to be both startling and disturbing, especially because of the decline in relative standing of U.S. students as they progress from elementary school through high school. This report describes how the TIMSS was conducted and discusses some lessons learned about the bases of these differences. The TIMSS included five main components: (1) curriculum analyses; (2) achievement tests; (3) questionnaire surveys of students, teachers, and administrators; (4) case studies of subjects in the United States, Germany, and Japan; the working environment and training of teachers; methods for dealing with differences in ability; and the role of school in adolescents' lives; and (5) a video study of classroom lessons in the United States, Germany, and Japan. The reports by members of the TIMSS staff express extreme caution in coming to firm answers concerning the poor performance of U.S. students. Nevertheless, it is possible to make some comments about American students' performance. Possible explanations begin with the fragmented, and nonsequential curricula in the United States, and the school's emphasis on developing rules that are automatically applied to problems rather than understanding the basis for the rules. Other problems are the lack of clear and tough standards, the mind-set that academic success is mostly determined by family background rather than by hard work, the demands placed on teachers, and their relatively low status within American culture. Demographic factors play a role, as does the associated phenomenon of placing some students in less challenging curricula. (Contains 6 tables, 4 figures, and 12 references.) (SLD)