Teachers' pedagogical reasoning and beliefs were examined by using questionnaires, observations and interviews. However, teacher's practical thinking, implicit pedagogical theory and beliefs are embedded in their practice (Shon, 1987 Elbaz, 2005). To compare teacher's pedagogy in four cultures (including USA), we use multi-voiced visual ethnography method (Tobin, 1991). This method treats video-clips as stimulus to elicit teacher's pedagogy. The video-clips of new type mathematical lessons in each culture are used.

Fifty-three teachers from eight elementary schools viewed five video clips of four countries, Singapore, Chinese, USA and Japan and answered by semi-structured interviews and short rating sheet. Japanese teachers found the differences between Singaporean, Chinese lesson and their own lessons as follows; First, Japanese teachers evaluated the high level problem Singapore and Chinese teachers used that children have to think deeply in lesson and they recognized the similarities of lesson styles in Singapore and Chinese. Second, most teachers found that speeds of lessons in both video-clips are faster than that of Japanese lessons. Third, they feel that teachers in two video clips lead and do not elicit various children's responses than that Japanese teachers supposed. Fourth, the methods of how teachers motivate children and give enjoyment of mathematics are different from some Japanese teacher's styles. The Singapore and Chinese video clips are not the representative lesson in each country but just as stimulus. These findings suggest that Japanese teachers take importance on quality of problems, lesson speeds, varieties of children's responses through classroom discourse, and relations between thinking and interestingness.

Questionnaire and interview methods are chosen to examine teacher’s educational ideas in four countries (Japan, Singapore, America and China) in the comparative research organized on the base of COE project from the University of Tokyo (as Prof. Kiyomi AKITA presents). Research in China shows that: (1) Chinese teachers show high appreciation of the new lessons characterized by problem-based inquiry, wide problem space, collective argumentation, and relaxed climate, especially new lessons form US and Japan. (2) Chinese teachers tend to be more conservative when they act as teachers or parents, which means that coverage of subject matter and students’ success in examination constitute their top concerns. Thus, in practice, they choose to act more as knowledge transmitter other than as organizers and facilitators of inquiry activities, as what their ideas and curriculum policy context might imply. (3) The contradiction and tense between Chinese teachers’ ideas and practices force Chinese teachers to seek more balanced and moderate ways to make teaching more effective. So, new Chinese lessons and Singaporean lessons are generally thought feasible, though they are not thought so problem-solving or inquiry-based. The feasible way seems to be traditional in instructional aim and activity structure but more engaging and active.

These findings suggest that reform in educational assessment, especially in external assessment, should try to make teachers feel that new way of teaching can be better in bring success in examination and that teachers need more professional support to design such new lessons.

International studies have revealed that East Asian students scored higher than those of other countries. In previous studies, however, the differences of mathematical thinking among East Asian students have not been seriously explored. This study examined the mathematical thinking of Japanese, Singaporean, and Chinese children by examining how they solved two types of problems: routine problems related to procedural knowledge (computation and routine word problems), and non-routine problems related to conceptual understanding. The latter problems required children to verbally express their thinking processes. Japanese, Singaporean, and Chinese fifth graders (n=1,199 in total) in public schools solved both the procedural and the conceptual problems. The children of three countries showed high level of performance on the procedural problems. The Chinese and Singaporean children scored higher than the Japanese ones. As for the conceptual problems, their overall performance was lower than that on the procedural problems. Chinese children scored higher than Japanese ones.As for their thinking processes involved in solving conceptual problems, differences among the three countries were prominent. Chinese children tended to rely on learned formula and preferred to use formal explanation. Singaaporean children tended to use various numerical knowledge to construct their solution strategies. The Japanese children tended to use everyday knowledge as well as numerical knowledge to construct their strategies, but some children avoided verbal explanation.Instructional materials and methods that reflect teachers’ beliefs about education in each country seem to influence not only students’ levels of procedural skills, but also their processes of mathematical thinking

This paper will examine the views, specifically the preferences of teachers, pupils and parents regarding the teaching and learning of Mathematics. Six primary schools in Singapore were involved in this study. After watching video excerpts of Singaporean and overseas teachers teaching Mathematics, the 90 respondents completed a survey questionnaire that sought their views. The groups of parents, teachers and pupils were then involved in a focus group discussion where they discussed specific topics about the Mathematic lessons, as well as teaching and learning processes. This paper will analyze their views related to the Mathematics lesson. The findings provide fascinating insights into the preferences and perspectives of the various groups regarding Mathematics lessons and teachers.