Analyses of both informal knowledge and cognitive barriers in ratio concepts

本研究旨在验证以下假设：分数中的内隐知识--等分和等整体概念的获得促进了分数的理解。为了验证这一假设，一个实验方案的基础上，均分区和平等的整体开发的三个三年级教室在日本的一所小学。作为对照，在两个三年级教室提供了基于日语教科书的传统教学。本研究测试的假设，给予这样的干预计划的学生将理解的顺序和幅度的分数比那些谁被指示使用传统的教科书更好的中心特征。实验组学生对分数等分和分数大小表征的理解明显好于课本组。此外，实验组和教科书组在教科书中经常发现的常规任务的表现上没有差异。这些结果进行了讨论，鉴于重要的教学目标，让学生理解分数。

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.