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This study is part of the international survey studies on teachers' beliefs concerning the integration of children's literature in mathematics teaching and learning, and this paper reports the findings of the thematic analysis of open-ended survey responses elicited from 287 primary school teachers and teacher trainees in Taiwan. Using the seminal social psychology theory, the Theory of Planned Behaviour (Ajzen in "Organizational Behavior and Human Decision Processes," "50," 179-211, 1991) to frame the findings, this study highlights 11 perceived barriers and 11 perceived enablers that are thought to influence the teachers' intention to integrate children's literature in their mathematics teaching. More specifically, we identified time constraint, lack of pedagogical knowledge and confidence, and resource constraint as being the most-cited perceived barriers, while pedagogical benefits, desire to improve teaching, and enabling social norms were identified as the top perceived enablers. Ultimately, this article offers several recommendations to address some of these key perceived barriers.

This study analysed the designs of algebraic problems in elementary mathematics textbooks used in Finland, Indonesia, Malaysia, Singapore, and Taiwan. This study employed a content-analysis method that focused on four dimensions: representational forms, cognitive demand levels (CDLs), contextual features, and types of responses. The result of the chi-square tests showed that there were significant differences between specific categories in the four dimensions among the five countries. Although we argued this, we did not claim that it might influence students' mathematics performance in these five countries. For instance, regarding representational forms, Finland, Indonesia, and Malaysia designed problems with the highest emphasis on symbolic forms (z = 21.62, 75.03%), visual forms (z = 15.65, 42.57%), and verbal (z = 10.69, 41.35%) forms, respectively. The other findings regarding CDLs, contextual features, and types of responses can also help to understand students' potential engagement when learning algebra at elementary schools in the five countries.

The purpose of this study is to compare the differences of four essential aspects (i.e., representational forms, contextual features, cognitive demand levels, and response types) embedded in mathematical problems within the topics of numbers and operations in mathematics textbooks used in Finland, Indonesia, Malaysia, Singapore, and Taiwan. This study employed a content analysis method to analyze the four aspects in the most representative textbook from each country. Results showed varied development of the four aspects and indicated different priorities being emphasized in each country's textbook. For instance, Singaporean textbooks have the highest percentage of mathematical problems that require higher levels of cognitive demands. Textbooks in Indonesia, Malaysia, Finnish, and Singapore contain more opportunities to engage with purely mathematical forms, with a percentage higher than 40%. Other findings concerning the four aspects can help to understand students' potential engagement with mathematics teaching and learning at elementary schools in the five countries.

Numbers and operations is widely considered the most essential domain in elementary school grades. Nevertheless, elementary students in Indonesia have consistently performed unsatisfactorily on this domain in several international mathematics assessments. This study examined 308 Indonesian sixth grade students' current performance(s), misconception(s), and confidence level(s) on a three-tier test containing 40 number sense-related questions. The results showed that most of the students performed unsatisfactorily (indicating low number sense) with a relatively high level of confidence and exhibited considerable misconceptions. The percentage of the correct responses (i.e. answers and reasons) was approximately 46.48% for the 40 questions. Specifically, only 25 students (8.12%) had good number sense, while 60.71% of the students held either strong or moderate misconceptions. The students' strong misconceptions were identified. In addition, other remarkable findings are presented and discussed. Furthermore, the implications of the results and the suggestions for future research are described.

The purposes of this study were to compare the differences in the use of geometry in elementary school mathematics textbooks among Finland, Mainland China, Singapore, Taiwan, and the USA and to investigate the relationships between the design of the textbooks and students' performance on large-scale tests such as TIMSS-4 geometry, TIMSS-8 geometry, and PISA space and shape. The content analysis method was used to collect data, and then chi-square tests and correlation analyses were used to analyze data. The results showed that there were significant differences in representation form, problem type, and question format among these mathematics textbooks from the five countries. Moreover, the strength of the positive relationships between visual form (combined form) and students' performance on TIMSS-4 geometry, TIMSS-8 geometry, and PISA space and shape decreases as students advance to higher grades, whereas increasing strength of correlations as students get older is found between contextual problems and students' performance on the three large-scale tests.

This study examined the performance of sixth graders in Hong Kong on a number sense three-tier test (NSTTT) and identified their possible misconceptions about number sense. The NSTTT comprises a content tier, which assesses the content knowledge of number sense; a reason tier, which assesses the reasons for the first-tier responses; and a confidence tier, which assesses students' confidence in their responses to questions in the first two tiers. The study participants were 942 sixth graders in Hong Kong. The results indicated that the percentage of correct answers (and confidence indices) of the first two tiers (i.e. the content tier and reason tier) of the sixth graders was 53.96% (4.27). In addition, the sixth graders showed the highest performance on "ability to recognise number size" and the lowest on "ability to judge the reasonableness of computational results". Moreover, the results revealed that some students showed a low performance on number sense but demonstrated extremely high confidence levels, indicating that they may have significant misconceptions regarding number sense and may lack of number sense. Furthermore, this paper discusses the educational implications of the findings.

Judging the reasonableness of computational results is pivotal for students to understand mathematical concepts. This domain is the most sensitive to the presence of misconceptions in mathematics. Confidence ratings can serve as a measure of the strength of students' conceptual understanding. This study investigated the performance, misconceptions, and confidence ratings of 942 Hong Kong sixth grade students when they were asked to judge the reasonableness of computational results. The results showed that the students performed unsatisfactorily at judging the reasonableness, with an average score of 3.45 (out of 8). Slightly more than half of the students (53.72%) selected the correct computational results, but more than 60% of those students could not judge the reasonableness of the computational results (49.71% had misconceptions and 11.24% simply guessed the answers). In addition, only 20.82% and 18.23% of the students could apply number-sense- and rule-based methods to judge the reasonableness, respectively. Moreover, only 5.73% of the students showed high performance with a high confidence rating, 3.18% exhibited low performance with a low confidence rating, and 35.46% of them showed low performance with a high confidence rating. Furthermore, this study discusses students' misconceptions, the implications of the study, and suggestions for future research.

When solving a mathematical problem, students who do not have sufficient conceptual understanding may perform poorly and exhibit misconceptions. This study was aimed to examine students' conceptual understanding and significant misconceptions when solving number sense-related problems. An online three-tier diagnostic test was administered to 125 fifth-grade students with varied socio-economic backgrounds in Hong Kong. Only 14.40% of the students exhibited high performance with high confidence, indicating that these students had a profound conceptual understanding of number sense. In addition, the majority of the students (66.40%) did not demonstrate number sense; these students exhibited several significant misconceptions and could solve the questions only by using a rule-based method or guessing. Accordingly, most students performed unsatisfactorily on number sense-related problems. This study is imperative in identifying early predictors and provides information for further compatible interventions in the teaching and learning of number sense in Hong Kong in particular and worldwide in general.

The purpose of this study was to use calculator-assisted instruction to help two fifth-grade low-achievers learn number sense. The research process includes three stages: (1) pre-test interview to detect what kinds of number sense the students did not have; (2) calculator-assisted instruction to help them develop number sense; and (3) post-test interview to examine if their number sense had improved. The results showed that students successfully developed two kinds of number sense abilities they did not have in the pre-test interview. However, they failed to develop the number sense ability of using a benchmark number to make an estimation. Generally, this study illustrated that these two low-achievers could learn number sense from calculator-assisted instruction.

This study aimed to examine the effects of remedial instruction on low-SES & low-math first graders' basic mathematics competence as well as their interest and confidence in mathematics learning. Fourteen participants of low-SES & low-math were selected from two classes totaling fifty-seven first graders at a public elementary school in central Taiwan. Results show that remedial instruction conducted during the study successfully improved the low-SES & low-math students' mathematics competence and enhanced their interest and confidence in mathematics. Remedial instruction in the form of story contexts conducted through small group collaboration with manipulatives appeared to improve low-SES & low-math students' mathematics learning. Implications related to remedial instruction for low-SES & low-math students are discussed.

To assess the strength of conceptual understanding of number sense, a three-tier number sense test (TTNST) for fifth-grade students was developed and validated in this study. The three-tier test includes a content tier that assesses content knowledge of number sense, a reason tier that assesses a reason given for selecting a first-tier response, and a confidence tier that assesses how confident fifth-grade students are in their responses in the first two tiers. A total of 819 fifth-grade students from elementary schools in Taiwan participated in this study. Collected data showed that the test had good reliability and validity. The results revealed that many of the sample students performed poorly on number sense but maintained extremely high confidence; this indicated that many of the students held severe misconceptions and some were lacking in number sense. In addition, this study confirmed that the inclusion of a third tier (with confidence ratings) in the number sense test can mitigate the weaknesses of a previously used two-tier test. The educational implications of the findings are discussed in this paper.

This study investigated the performance of 790 fourth graders from Taiwan for judging the reasonableness of a computational result by using corresponding methods pertaining to whole numbers. A number-sense web-based two-tier diagnostic test was used to assess the number-sense performance of the fourth graders who participated in this study. The results revealed unsatisfactory performance of study participants. Only 25% of them could apply the number-sense-based method to answer problems, and approximately 38% of them had misconceptions regarding various items. This paper discusses several of these misconceptions, as well as the study implications and suggestions for future research.

This study analyzed how the earliest teaching and learning of probability was organized in Singapore, the US, and Indonesia through an analysis of textbook as an intended mathematics curriculum. An analytical framework was developed comprising four perspectives: representational forms, cognitive demand levels, contextual features, and organization of topics and its uniqueness. This study disclosed that the organization of the earliest teaching and learning of probability in the three countries focused on approximately similar topics, such as basic concepts of probability and probability measurements, although they were intended for different educational levels. Nevertheless, the uniqueness of the organization indicated that students in the three countries may receive different opportunity-to-learn in the earliest teaching and learning of probability in schools. In addition, Singaporean textbook had more exposure to higher-cognitive demand levels and written forms, while the US textbook put more concerns on lower-level demands and visual forms, Indonesian textbook included more proportions of lower- cognitive demand levels and written forms. Moreover, the three textbooks provided considerable opportunity to engage in real-life situations. Considering these results, more studies analyzing other standards-based textbooks and teaching practices might be needed to have further understanding of the organization of probability teaching and learning in schools.

The key purpose of this study was how we taught the use of benchmark strategy when comparing fraction for fifth-graders in Taiwan. 26 fifth graders from a public elementary in south Taiwan were selected to join this study. Results of this case study showed that students had a much progress on the use of benchmark strategy when comparing fraction after the experiment. This indicated that an experienced teacher played a key role and knew how to help their children lean the use of benchmark appropriately when comparing fraction.

Two 12-question estimation instruments were designed to compare the differences of estimating strategies used by the 8th-graders when solving contextual and numerical problems. Both instruments are parallel, meaning that the numbers used in both instruments are the same; however, they were presented differently. One hundred and ninety-eight 8th-graders in southern Taiwan were selected to participate in this study. The correct percentage for the numerical problems and contextual problems were 45% and 33%, respectively. The t-test result shows a significant difference for the estimation performance between contextual and numerical problems at alpha = 0.05. This implies that these 8th-graders perform significantly better on numerical problems than that of contextual problems. In addition, the t-test results show that there is a significant difference in the performance of reformulation, transformation, and compensation dimensions for numerical and contextual problems at alpha = 0.05, respectively. The results show that students performed better on numerical problems than on contextual problems for each dimension. In addition, data show that students tended to use the written method to solve numerical and contextual problems instead of estimation. Implications are discussed. (Contains 3 tables.)

This study examines the different ways in which fractions are dealt with in two elementary mathematics textbook series: Kang Hsuan (KH) in Taiwan and Laskutaito in Finland. The results showed that the total number of fraction questions in Laskutaito is higher than that in KH and that Laskutaito highlights the basic definition of fractions to help children develop higher levels of reasoning. Fraction questions in Laskutaito were presented with a number of pictorial representations to clarify the concepts. Laskutaito includes more symbolic representation of questions. Moreover, the questions posed in the Laskutaito reflect real-life situations, whereas the questions posed in KH do not connect with real life. The implications of this study are discussed and suggestions for future studies are made in this paper.

This study analyzed geometry problems in four middle-grade mathematics textbook series from Taiwan, Singapore, Finland, and the United States, while exploring the expectations for students' learning experiences with these problems. An analytical framework developed for mathematics textbook problem analysis had three dimensions: representation forms, contextual features, and response types. The results showed that the Taiwanese and Singaporean textbooks contained more problems in combined form, whereas the Finnish and American textbooks contained more problems in verbal and visual forms. The problem distribution across various representation forms was more balanced in the Finnish and Singaporean textbooks than in the Taiwanese and American textbooks. Most problems were non-application and close-ended problems compared to other application and open-ended problems. The Taiwanese textbooks contained the lowest proportion of real-world problems, whereas the American textbooks contained the highest proportion of open-ended problems. Implications of this study's findings for textbook developers and future research directions are discussed.

Organization for Economic Co-operation and Development (OECD) (2016) released the results of Programme for International Student Assessment (PISA) 2015 and reported that the students' performance in mathematics of Singapore and Indonesia had significant differences. There is a strong relationship between textbooks used and mathematics performance of the students. If textbooks differ, students will get a different opportunity to learn and the opportunity to learn influences students' achievement (performance). The purpose of this study was to analyze the trigonometry contents and cognitive demand levels in Singaporean and Indonesian mathematics textbooks. The data were primarily qualitative. Horizontal and vertical analyses were used in this study. The result showed that Singaporean textbooks put more emphasis on all the concepts of trigonometry on right-triangle and further trigonometry (sine/cosine rules) while Indonesian textbooks provided more discussions on angle and its concepts, trigonometry on right-triangle, and graph function of trigonometry (which is a lot more difficult than sine/cosine rules). In addition, Singaporean textbooks provided more mathematical questions requiring higher cognitive demand levels while, Indonesian textbooks provided more questions requiring lower levels. The differences of textbooks' contents and required cognitive demand level probably influenced students' mathematics performance in the two countries. It is hoped that the results will inform curriculum designers and/or textbooks' author(s) in Indonesia, Singapore, and other countries as they review and update the mathematics curriculum and/or mathematics textbooks.

Most students thinking mathematics is a difficult subject. This study aims to enhance students' motivation and efficiency in learning mathematics. This study developed 3D animation on the binomial theorem with historical stories of mathematics as the plot. It also examined the effect of animation on students' learning willingness and comprehension. And using questionnaire to collect data. The results showed that enhance the students' comprehension of the binomial theorem, suggesting that 3D animation could effectively improve mathematical learning. Moreover, the results showed that the students' willingness to learn mathematics also increased, suggesting that 3D animation can be used as an auxiliary learning tool for students. The results demonstrated that using 3D animation can improve students' learning willingness, comprehension, and application of mathematics.

A web-based two-tier test (WTTT-NS) which combined the advantages of traditional written tests and interviews in assessing number sense was developed and applied to assess students' answers and reasons for the questions. In addition, students' major misconceptions can be detected. A total of 1,248 sixth graders in Taiwan were selected to participate in this study. Results showed that the average percentage of correct answers was about 45%. Among the students who chose correct answers, about 22.9% of them used a number sense method to solve problems. In addition, students' misconceptions are classified by content domains. The major contributions of the WTTT-NS are to (1) avoid students using written computations answering number sense questions; (2) present a whole picture of students' misconceptions and the weight of these misconceptions; (3) include the strengths of quantitative and qualitative methods; (4) identify students' "true understanding" (a correct answer based on their correct understanding instead of guessing) by exploring reasons for their choices. In sum, the WTTT-NS is a new worthwhile method to assess students' number sense competence.

The purpose of this study was to examine the differences between Finnish and Taiwanese textbooks for grades 7 to 9 on the topic of solving systems of linear equations (simultaneous equations). The specific textbooks examined were TK in Taiwan and FL in Finland. The content analysis method was used to examine (a) the teaching sequence, (b) application types, (c) representation forms, (d) response types, and (e) level of cognitive demand. The main difference between the Finnish and Taiwanese textbooks was that the Finnish textbooks introduced the topic of linear systems using a graphical approach, while the Taiwanese textbooks used an algebraic approach. Results also showed that the Taiwanese textbooks had fewer problems, but more challenging problems requiring a higher level of cognitive demand; the Finnish textbooks had more authentic application problems, and even more problems were displayed in visual forms. In addition, the Taiwanese textbooks had more open-ended problems, particularly problems asking students to explore or explain, whereas the Finnish textbooks did not have exploration problems. The topic of linear systems was also taught earlier in Taiwanese textbooks. Implications of this study and suggestions for future studies are discussed.

To examine the relative performance in number sense among low-SES students, new immigrant students, and typical learners in grades 4 through 6, data were collected through a number sense web-based, two-tier test. A total of 628 fourth graders, 535 fifth graders, and 524 sixth graders in Taiwan participated in this study. Results showed that there were statistically significant differences in number sense performance among new immigrant children, low-SES students, and typical learners in the fourth, fifth, and sixth grades. The subsequent post hoc comparisons indicated that there was a statistically significant difference between typical learners and new immigrant children in fourth and fifth grades. Moreover, there were statistically significant differences between typical learners and low-SES students in fifth and sixth grades. The Chi-squared test results also showed that there were significant differences in the uses of solution methods between the fourth, fifth, and sixth graders. Implications of this study and suggestions for the future studies are discussed.

Background: Number sense is a key topic in mathematics education, and the identification of children's misconceptions about number is, therefore, important. Information about students' serious misconceptions can be quite significant for teachers, allowing them to change their teaching plans to help children overcome these misconceptions. In science education, interest in children's alternative conceptions has led to the development of three- and four-tier tests that not only assess children's understandings and misconceptions, but also examine children's confidence in their responses. However, there are few such tests related to mathematical content, especially in studies of number sense. Purpose: The purpose of this study was to investigate children's performance and misconceptions with respect to number sense via a four-tier diagnostic test (Answer Tier ? Confidence rating for Answer Tier ? Reason Tier ? Confidence rating for Reason Tier). Design and method: A total of 195 fifth graders (10-11 years old) from Taiwan participated in this study. The four-tier test was web-based and contained 40 items across five components of number sense. Findings: The results show that (1) students' mean confidence rating for the answer tier was significantly higher than for the reason tier; (2) an average of 68% of students tended to have equal confidence ratings in both answer and reason tiers; (3) students who chose correct answers or reasons had higher mean confidence ratings in most items (36 out of 40) than those who did not; and (4) 16 misconceptions were identified and most of them were at a strong level. Conclusion: The four-tier test was able to identify several misconceptions in both the answer and reason tier and provide information about the confidence levels. By using such information, teachers may be better positioned to understand the nature of learners' misconceptions about number sense and therefore support their pupils' progress in mathematics.

The authors compared the mental computation performance and mental strategies used by an experimental Grade 2 class and a control Grade 2 class before and after instructional intervention. Results indicate that students in the experimental group had better performance on mental computation. The use of mental strategies (counting, separation, aggregation, and holistic) and atypical strategies in the control group decreased or even disappeared after the intervention. However, students in the experimental group could apply higher level mental strategies, such as aggregation, holistic, short-jump, and atypical strategies after the intervention. These findings indicate that the teaching of the vertical addition/subtraction algorithm does not promote the development of mental computation.

This study examined (a) the differences in preservice teachers’ procedural knowledge in four areas of fraction operations in Taiwan and the United States, (b) the differences in preservice teachers’ conceptual knowledge in four areas of fraction operations in Taiwan and the United States, and (c) correlation in preservice teachers’ conceptual knowledge and procedural knowledge of fractions in Taiwan and the United States. Participants were preservice teachers ("N" = 49) in a teacher education program in the United States and comparable Chinese preservice teachers ("N" = 47). Results indicated that Chinese preservice teachers performed better in procedural knowledge on fraction operations than American preservice teachers. No significant differences were found for conceptual knowledge on fraction division. Further, the correlation in this study showed that for Chinese and American preservice teachers, the relationship between conceptual and procedural knowledge of fraction operations was weak. (Contains 4 tables and 1 figure.)

The purpose of this study was to examine the relative effectiveness of two different learning modes; namely, a computer animation self-directed learning approach and a paper version of the self-directed learning approach, to 5th-graders' number sense development. Two 5th-grade classes, 30 students each, were selected from a public elementary school in southern Taiwan and randomly assigned to the computer animation experimental group [CAEG] and the paper-version group [PAG]. The major finding indicated that students in the CAEG group had better performance on number sense and showed more frequent uses of number sense than students in the PAG group. Implications for the uses of computer animation self-directed learning activities are discussed.

This study investigated and compared the geometry knowledge and levels of pre-service elementary teachers from the United States and Taiwan. Forty pre-service teachers in Taiwan and 48 pre-service teachers in the United States at the beginning of their teacher education programs completed the Entering Geometry Test (EGT) and the van Hiele Geometry Test (VHGT) developed by Usiskin (1982). The data gathered through these two tests was analysed to determine if there were any significant differences in the geometric knowledge and van Hiele levels between the American and Taiwanese pre-service elementary teachers. The findings indicated that Taiwanese pre-service elementary teachers had significantly higher mean scores on the EGT and achieved higher van Hiele levels than their U.S. counterparts. Possible factors that contribute to these differences as well as the implications are discussed in the paper. (Contains 7 tables.)

To investigate the structure of number sense and then to assess its uses in fifth-grade children's number sense development, a computerized number sense scale was developed and evaluated. The findings of the study indicate that the newly developed scale, with four dominant factors identified and reconfirmed, is internally consistent and substantially valid. It can be reliably used as a screening measure for a quick check of students' number sense development via online self-assessment. Compared with our previous study, both qualitative and quantitative changes were detected in students' number sense development at different grades. The qualitative change in number sense development is manifested in different numbers of factor components produced at different grades. The quantitative change is manifested in different amounts of factor variance explained at different grades. Furthermore, among the four aspects of number sense, Taiwanese students perform best on recognizing the relative number size yet relatively worse on judging the reasonableness of computational results. (Contains 4 tables.)

This research examined the presentation of fractions in textbooks used by fifth and sixth graders in Singapore, Taiwan, and the United States. The specific textbooks examined were "My Pals Are Here! Maths" (MPHM) in Singapore; "Kung Hsung" (KH) in Taiwan; and "Mathematics in Context" (MiC) in the USA. Results show the problems posed in MiC put more emphasis on real-life situations than KH textbooks in Taiwan and MPHM in Singapore. Designing materials that provide opportunities to connect mathematics content with applications in real life is consistent with recommendations from professional organizations. The activities in KH and MPHM tended to emphasize procedures, while the activities of MiC focused more on conceptual understanding and less on the development of procedures. An examination of the mathematics textbooks revealed that MPHM introduced and developed fractions the earliest among the three countries investigated and the content taught in MPHM was about one grade earlier than when the same content was experienced by students in KH and MiC. (Contains 3 tables and 3 figures.)

The authors compared the performance of students who received integration of number sense activities in instruction with students who received instruction using regular mathematics textbooks. Two classes of third-grade students (N = 60) were randomly assigned to experimental and control groups. Students in each group were given a pretest, post I test, and post II test, and were interviewed individually across 20 class periods of intervention. Results showed that the performance of the students in the experimental group was significantly higher than those in the control group. The interview data also showed that the students in the experimental group performed better than students in the control group in flexible and effective ways to use number sense. (Contains 3 figures and 5 tables.)

A quasi-experimental design was adopted to investigate the effect of integrating technology into mathematics teaching on students' number sense and their learning attitudes. Two sixth-grade classes were selected from an elementary school in Taiwan for participation in this study. The control group with 32 students followed their usual mathematics instruction on number sense without using technology. The experimental group with 32 students learned number sense in a technology-based environment. The ANCOVA results showed a statistically significant difference between the control group and the experimental group on number sense performance. The "t"- test results showed no significant difference before and after the instruction for the control group. However, there was a statistically significant difference before and after the instruction for the experimental group. Furthermore, data also showed a significant difference in the students' learning attitudes between the control group and the experimental group after the teaching. These results indicated that integrating technology into number sense teaching and learning not only promote students' number sense, but also has a positive effect on attitudes towards learning number sense. (Contains 5 tables and 6 figures.)

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.

The main purpose of this study was to investigate the number sense performance of 3rd-graders in Taiwan, and to diagnose areas of weakness or deficiency in number sense development. A total of 808 3rd-graders participated in this study. The results indicated that these students did not perform well on each of the five number sense components (correct rates approx. 34%), and they appeared worst on the performance of "Judging the reasonableness of computational results". Boys and girls did not show any appreciable difference in their ability to solve number sense problems. The importance of number sense should be highlighted both by teachers and in textbooks and more time and opportunity provided for students to work on this type of exercise at lower grade levels. This would require that "drill and practice" exercises in mathematics should not indeed be over-taught, and the teaching of number sense to children should begin as early as possible. (Contains 8 tables.)

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 "recognizing relative number size" and perform worst on "judging 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 the strategies used by pre-service teachers when responding to number sense related questions. 15 pre-service teachers from one University in Southern Taiwan were interviewed. Results indicated that about one-third of these pre-service teachers were able to use number sense strategies (such as understanding the meaning of numbers, operations and their relationships, recognizing the number size, developing and using benchmarks appropriately, and judging the reasonableness of a computational result by using the strategies of estimation) and the other two-thirds' relied heavily on written algorithms to solve problems. This is consistent with the findings of the earlier studies (Reys & Yang, 1998: Yang & Reys, 2002; Yang, 2003), which state that fifth, sixth and eighth graders in Taiwan rely heavily on the written method when responding to number sense related questions. This implies that the performance of pre-service elementary teachers on number sense is low. Furthermore, the results of this study also support that "effective teaching requires knowledge and understanding of mathematics" and the reports of the earlier studies and documents (Ma, 1999; NCTM, 1991, 2000; Schifter, 1999) that teachers need a profound understanding of important mathematical concepts and must "be able to represent mathematics as a coherent and connected enterprise" and "know and understand deeply the mathematics they are teaching and be able to draw on that knowledge with flexibility in their teaching tasks." (Contains 2 tables and 1 figure.)

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.