“中国”“新加坡”“英国”教材中分数除法学习路径的比较研究

A Comparative Study on the Learning Trajectories of Division with Fractions in Textbooks from China,Singapore and the UK

以中国人教版小学数学教材、英国SMP版小数数学教材和新加坡MC版小学数学教材为研究对象,比较了3国教材中分数除法的学习路径.结果表明:3国均按照"分数除以整数→整数除以分数→分数除以分数"的顺序来安排学习任务;分数除以整数的任务均采用了等分除的模型,整数除以分数、分数除以分数(国内统称为一个数除以分数)的任务大都采用了包含除模型;任务的表征方式既有直观表征,也有抽象表征.新加坡教材中的学习路径所包含的任务最详细,其推进过程可谓小坡度、慢步走;中国教材的学习路径所包含的任务最简略,其推进过程可谓大跨度、快步跑;英国教材的学习任务最为抽象.最后基于分析,重构了分数除法的学习路径.

In this paper,we compare the learning trajectories of the division of fractions in Chinese elementary school mathematics textbooks,the British SMP school mathematics textbooks,and the Singaporean MC primary school mathematics textbooks.We found that the learning tasks are arranged in the order of"dividing a fraction by a whole number→dividing a whole number by a fraction→dividing a fraction by a fraction."The tasks of dividing a fraction by a whole number all adopt the portative models,and the tasks of dividing a whole number by a fraction and dividing a fraction by a fraction(generally referred to as dividing a number by a fraction in China)mostly adopt the quotitive models.The representations of tasks included both visual and abstract representations.The learning trajectories in the Singaporean textbooks contain the most detailed tasks,which could be described as proceeding in small steps and at a slow pace.The Chinese textbooks contain the lowest number of tasks,which could be described as proceeding in big steps and at a fast pace.The British textbooks contain the most abstract tasks because they were all purely mathematical situations.Based on these analyses,we propose the hypothetical learning trajectories of the division of fractions.