Analyses of both informal knowledge and cognitive barriers in ratio concepts

本研究旨在检验分数中隐含知识的等分割和等整体概念的习得是否促进了分数的理解。为了验证这一假设，在日本一所小学的三个三年级教室中开发了一个基于等分割和等整体的实验程序。作为对照，在两个三年级的教室中提供了基于日语教科书的传统教学。目前的研究测试了这样一个假设，即接受这种干预计划的学生比那些接受传统教科书指导的学生更能理解分数的顺序和大小作为核心特征。实验组的学生比教科书组的学生对分数的等分和分数大小的表示有更好的理解。此外，在教科书中经常出现的日常任务的表现上，实验组和教科书组之间没有差异。鉴于让学生理解分数的重要教学目的，我们讨论了这些结果。

The present study was designed to test the hypothesis that acquisition of the concepts of equal-partitioning and equal-whole, which constitute implicit knowledge in fractions, promotes understanding of fractions. To test the hypothesis an experimental program based on both the equal-partitioning and equal-whole was developed for three third grade classrooms in a Japanese elementary school. As a control, conventional teaching based on a Japanese textbook was provided in two third grade classrooms. The present study tested the hypothesis that students given such an intervention program would understand order and magnitude as central characteristics of fractions better than those who had been instructed using a traditional textbook. Students in the experimental group showed significantly better understanding of the equal-partitioning of fractions and the representation of the sizes of fractions than did those in the textbook group. There were, moreover, no differences between the experimental and textbook groups in the performance of routine tasks frequently found in the textbook. These results are discussed in view of the important instructional aims of having students understand fractions.