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S. students. S. S had fewer school days per year (178) than China (251), Korea (222), and Taiwan (222). , and Taiwan, ranging from 1 044 to 1 162 hours per year. By the fifth grade, however, the differences were more pronounced: fifth-grade children attended school for 1 044 hours per year in the U S . , 1 655 hours in Taiwan, and 1 466 hours in Japan. Thus, compared to Korea, Japan, China, and Taiwan, the United States has the shortest school year. S. schools devote the least amount of school time to mathematics.

332) These previous investigations demonstrated the value of error analysis in capturing students' understanding of mathematical knowledge. The examination of students' misconceptions provides an indication of their proficiencies in mathematical problem solving and reasoning. Mathematical justz$cation. 29), "make and evaluate mathematical conjectures and arguments," and "validate their own thinking" (p. 81). Studies by Senk (1985, 1989) showed not only that students have great difficulty in providing sound mathematical justifications when solving geometric proof problems but also that the degree of difficulty was related to students' levels of geometric thought.

1989) replicated the Carraher et al. S. students as subjects. Interestingly, they did not find a significant effect for context (symbolic computation problems vs. textbook-like word problems vs. situation problems). S. and Brazilian students on the symbolic computation tasks is larger than the difference on the textbook-like word problems and situation problems. S. and Brazilian students may be derived differently, depending on the tasks the comparison is based on. Thus, the nature of the tasks is a crucially important factor to consider in measuring students' mathematical performance.

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