Effect of Instruction Focusing on Conceptual Understanding of Calculation

本研究的目的不仅在于阐明三至五年级小学生的概念知识与四种算术运算规则(加、减、乘、除)之间的关系，而且考察四种算术运算的教学效果。研究得出了以下5点结论：1.对减法、乘法、除法等问答方法的理解存在显著的相关关系。“计算发明”任务的解题水平与计算的检查-回答方法显著相关。不熟悉的“计算发明”任务要求儿童用加法计算乘法，用减法计算除法。作为方差分析的结果，我们观察到对“借来的号码到哪里去了？”的回答有所改善。带减法，并根据孩子的年龄“发明乘法”。理解减法的含义，例如“你从高阶数借了多少个数？”，“借来的数去了哪里？”，以及“消失的数是什么数？”与“考答法”和“计算的发明”有着密切的关系。研究结果表明，对减法意义的清晰理解是后续乘除概念理解的基础。让孩子们思考计算意义的指导在调查后不久进行的测试中取得了良好的结果。但一年后的测试显示，教学效果开始下降。这表明，在学校教育中反复体验思考计算的意义对于获得对计算的准确概念理解是必要的。

The purpose of this research was not only to clarify the relation of third- to fifth-graders' conceptual knowledge among four rules of arithmetical calculation (addition, subtraction, multiplication and division) but also to examine the effect of instruction which made children think about the meaning of four calculations. The following 5 points emerged from the research.1. Meaningful correlation was observed among the understanding of checking-answer methods of subtraction, multiplication and division.2. Significant correlation was found between the solution level of "invention of calculation" task and checking-answer method of calculations. Unfamiliar "invention of calculation" task required the children to solve multiplication by using addition and to solve division by using subtraction.3. As the result of ANOVA, we observed some improvement in the response to "where did the borrowed number go?" with subtraction, and "invention of multiplication" according to the children's age.4. Understanding of the meaning of subtraction such as "how many numbers did you borrow from upper rank number?", "where did the borrowed number go?", and "what number was the gone number?" was deeply related with "checking-answer method" and "invention of calculation". It was shown that clear understanding of the meaning of subtraction lie at the base of conceptual understanding of multiplication and division afterwards.5. The instruction that made children think about the meaning of calculations brought good results in the test given soon after the investigation. But the test given in a year later showed the effect of instruction came to decline. It suggested that repeated experience to consider the meaning of calculation in school education is necessary for acquisition of precise conceptual understanding of calculations.