This study shows that Singaporean future teachers are weaker in the competence of devising formal mathematical arguments and transforming heuristic ideas into valid proofs than Taiwanese future teachers. The relative strengths and weaknesses of these two countries, as compared to other TEDS-M higher-achieving Western countries, such as Germany, Poland, Russia, Switzerland, and the USA, are examined after reselecting and reclassifying the achievement items based on Niss’ (2003) mathematics competence and Hsieh’s (Secondary Education, 63(3), 2012) mathematics teaching competence structures. Taiwan and Singapore are identified as the only two Asian countries integrating the Chinese/Confucian tradition in their education regarding TEDS-M. This study investigates the similarities and differences of mathematics-related teaching competencies between the future secondary school teachers of Taiwan and Singapore by using data from the international Teacher Education and Development Study in Mathematics (TEDS-M), organized by the International Association for the Evaluation of Educational Achievement. There is strong evidence that differential performance between Western and Asian countries in PISA and TIMSS can be directly attributed to the types of items in the respective tests. These strengths and weaknesses are then linked back to the contents of the PISA and TIMSS tests. In this way, the relative strengths and weaknesses of Western and Asian countries are identified. Differential performances of six countries on each set of items are examined. In particular, many geometry and algebra items in TIMSS are “inner mathematics” (mathematics without a real-life context), and such items do not appear in the PISA test. In this paper, TIMSS released items are divided into two sets: one that fits the PISA framework and one that does not. It was found that Western countries generally performed better in PISA than in TIMSS, and Eastern European and Asian countries generally performed better in TIMSS than in PISA.
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