This study used a diagnostic testing approach to compare the mathematics achievement of eighth-grade students across a sample of 20 countries, analyzing data from the Third International Math and Science Study-Revised (TIMSS-R, 1999). Using the rule-space method, student mastery was measured on 23 specific content knowledge and processing subskills ("attributes") underlying students' item scores, using 23 attributes previously defined and validated. Mean mastery levels for each attribute were compared for the 20 selected countries. Clear differences among the countries were found in patterns of subskill achievement. U.S. students were strong in some content and quantitative reading skills, but weak in others, notably geometry. Interestingly, success in geometry was found to be highly associated with logical reasoning and other important mathematical thinking skills across the sampled countries. (Contains 4 tables, 8 figures, and 1 note.)