Gilmore Camilla K, Bryant Peter
University of Oxford, UK.
Br J Educ Psychol. 2006 Jun;76(Pt 2):309-31. doi: 10.1348/000709905X39125.
Background and aims. In order to develop arithmetic expertise, children must understand arithmetic principles, such as the inverse relationship between addition and subtraction, in addition to learning calculation skills. We report two experiments that investigate children's understanding of the principle of inversion and the relationship between their conceptual understanding and arithmetical skills.
A group of 127 children from primary schools took part in the study. The children were from 2 age groups (6-7 and 8-9 years).
Children's accuracy on inverse and control problems in a variety of presentation formats and in canonical and non-canonical forms was measured. Tests of general arithmetic ability were also administered.
Children consistently performed better on inverse than control problems, which indicates that they could make use of the inverse principle. Presentation format affected performance: picture presentation allowed children to apply their conceptual understanding flexibly regardless of the problem type, while word problems restricted their ability to use their conceptual knowledge. Cluster analyses revealed three subgroups with different profiles of conceptual understanding and arithmetical skill. Children in the 'high ability' and 'low ability' groups showed conceptual understanding that was in-line with their arithmetical skill, whilst a 3rd group of children had more advanced conceptual understanding than arithmetical skill.
The three subgroups may represent different points along a single developmental path or distinct developmental paths. The discovery of the existence of the three groups has important consequences for education. It demonstrates the importance of considering the pattern of individual children's conceptual understanding and problem-solving skills.
背景与目的。为了培养算术专长，儿童除了学习计算技能外，还必须理解算术原理，例如加减法之间的逆关系。我们报告了两项实验，研究儿童对逆运算原理的理解以及他们的概念理解与算术技能之间的关系。
一组来自小学的127名儿童参与了该研究。这些儿童来自两个年龄组（6至7岁和8至9岁）。
测量儿童在各种呈现形式以及标准形式和非标准形式下解决逆运算问题和对照问题的准确性。还进行了一般算术能力测试。
儿童在逆运算问题上的表现始终优于对照问题，这表明他们能够运用逆运算原理。呈现形式影响表现：图片呈现使儿童能够灵活运用他们的概念理解，而不管问题类型如何，而文字问题则限制了他们运用概念知识的能力。聚类分析揭示了三个具有不同概念理解和算术技能特征的亚组。“高能力”组和“低能力”组的儿童表现出与其算术技能相符的概念理解，而第三组儿童的概念理解比算术技能更先进。
这三个亚组可能代表了单一发展路径上的不同点或不同的发展路径。发现这三个组的存在对教育具有重要意义。它表明了考虑个体儿童概念理解和解决问题技能模式的重要性。
