Effect of Instruction Focusing on Conceptual Understanding of Calculation

本研究的目的，不仅在于厘清三至五年级学童对四种算术运算规则（加减乘除）的概念性知识的关系，也在于检视让学童思考四种运算规则意义的教学效果。研究得出以下5点结论：1.研究结果表明：1.小学生对减法、乘法和除法的理解存在显著相关.“计算发明”任务的解决水平与计算的检查-回答方法之间存在显著相关。不熟悉的“发明计算”任务要求幼儿用加法解乘法，用减法解除法.作为方差分析的结果，我们观察到在回答“借来的数字去哪里了？”根据孩子的年龄，用减法和“发明乘法”。理解减法的意义，如“你从上位数中借了几个数？”“，“借来的号码去哪儿了？“，以及“消失的号码是多少？与“对答法”、“发明计算”有着密切的关系。研究表明，对减法意义的清晰理解是对乘除概念理解的基础.让孩子们思考计算的意义的指导在调查后不久的测试中取得了良好的结果。但一年后的测试表明，教学效果开始下降。它建议在学校教育中反复考虑计算的意义是必要的，以获得精确的概念理解的计算。

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.