In an interview setting, 21 6th-graders from four public schools in south Taiwan were asked a series of questions, related to whole and decimal numbers, designed to assess their number sense. Eight students were randomly selected from both low and middle performance levels, and five were selected from a high-performance level, according to their mathematical capabilities. Results indicated that, regardless of performance level, very few number sense strategies (e.g. using benchmarks, estimation or numbers of magnitude) were used. The evidence also revealed that Taiwanese students tended to apply rule-based methods and standard written algorithms to explain their reasoning. Their inclination to use paper-and-pencil procedures narrowed their thinking and reasoning powers: this heavy reliance on written algorithms seemed to be a major impediment to the development of number sense. RationaleThe recent Third International Mathematics and Science Study (Mullis et al., 2004) confirmed the evidence from previous research that Taiwanese students consistently achieve high marks in mathematics performance, involving written computation (Stevenson et al., 1986;Stigler et al., 1991;Stevenson et al., 1993;Beaton et al., 1997). However, McIntosh et al. (1992) claim that students who are highly skilled in paper-and-pencil computation (often the gauge by which success in mathematics is measured) may or may not have developed good number sense. The study of Reys & Yang (1998) confirmed that Taiwanese students' proficiency, in exact written computational performance, is not necessarily accompanied by good number sense. The Israeli teacher Muneer (1999) repeated this study with 9th-grade students and

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In order to investigate the performance of number sense and its relationships with mathematics achievement of Taiwanese students who had just completed the 5th-grade mathematics curriculum, a computerized number sense scale has been developed. This number sense scale includes four factors which are recognizing relative number size, using multiple representations of numbers and operations, judging the reasonableness of estimates of computed results and recognizing the relative effect of operations on numbers. A total of 1,212 students in Taiwan participated in this study. The main findings of this study are summarized as follows. First, the students perform best on Brecognizing relative number size^and perform worst on Bjudging the reasonableness of estimates of computed results^. This finding is consistent with previous studies. It shows that students in Taiwan seem quite poor on judging the reasonableness of estimates of computed results. Second, female students, on average, have higher scores on recognizing the relative number size than male students, even though only a small effect size is found. And, third, the achievements of the students in mathematics are significantly correlated with their number sense, as measured by the average grade for the academic year of 5th-grade students.

This study examined number sense strategies and misconceptions of 280 Taiwanese pre-service elementary teachers who responded to a series of real-life problems. About one-fifth of the pre-service teachers applied number sense-based strategies (such as using benchmarks appropriately or recognizing the number magnitude) while a majority of pre-service teachers relied on rule-based methods. This finding is consistent with earlier studies in Taiwan that fifth, sixth, and eighth grade students tended to rely heavily on written methods rather than using number sense-based strategies. This study documents that the performance of pre-service elementary teachers on number sense is low. If we want to improve elementary students' knowledge and use of number sense, then action should be taken to improve the level of their future teachers' number sense.

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