分数がわからない大学生に接してから疑問に思い、分数の導入学年や指導時間について昭和22年から現在まで調べてみると、分数の指導内容は変わらないのに、分数導入学年が一貫せず、指導学年数も分数の指導時間数も文部科学省の学習指導要領の改定ごとにかなり減少していることが明らかになった。分数導入学年が一貫していないのは、分数導入の基本である分割行動の子どもの発達に関する研究がほとんどなされていないためである。分数の基本となる子どもの分割行動を研究し、数学教育の視座から子どもの発達に応じた分数導入時期を検討することをテーマとした。先行研究のように人形や皿のような配分先を指定せず、「同じに分ける」事を教示しないで4歳児から7歳児80名について連続量と離散量についての分割概念を調べる実験を行った。実験の結果、連続量と離散量の等分割行動の発達が異なることを明らかにした。さらに、子どもの分割行動の発達過程の分析を試みた。そして、子どもの分割行動の発達に応じた低学年での分数導入時期を提案した。
Development of Partitioning Skills: A proposal for teaching fractions
When introducing any new mathematical skill to children, it is important to consider their cognitive development. Therefore, in teaching fractions to children it is necessary to have a clear understanding of how young children develop partitioning skills. Most previous studies on the development of partitioning skills specified objects such as dolls or plates. Sometimes, children were explicitly instructed to "divide items in equal quantities". Because allotting items to objects or placing them on plates can be achieved on the principle of one-to-one correspondence, successful performance of these tasks does not necessarily reflect true understanding of the principles of partitioning. Therefore, I studied the development of partitioning skills among young children without specifying objects for partitioning and without instructing them to "divide equally."
In the experiment, incorrect solutions were first shown to children and they were asked whether the solutions were the right ones to test childrenﾕs understanding of the basic principles involved in partitioning. If children rejected the solutions, they were then encouraged to solve the problems by themselves. Based on preliminary observations, two types of partitioning problems were designed: partitioning of continuous (dividing a lump of clay) and discrete (marbles) quantities.
Testing 80 young children of the age from four to seven years, different developmental trends were found for discrete and continuous quantities. Children showed marked increase in the understanding of the principles of partitioning for continuous quantity from five to six years old. In contrast, their development of partitioning skills for discrete quantities accelerated between the age of six and seven. Based on these findings stages of partitioning skill development were proposed and an optimal time for teaching fraction skills was specified. Presently in Japan, fractions are introduced in the final semester of the fourth grade. However, as 83% of the second graders have achieved the understanding of equal division of continuous quantities, fractions of continuous quantities may be introduced to the second graders without much cognitive burden.