美國國小兒童對於分數大小之理解及其思考策略

A Study of American Children’s Understanding of Fraction Size and Their Thinking Strategies

培養學生正確的數量感（number sense）是數學教學的重要目標之一。然而，許多研究指出：國小學生的數量感遠落於運算能力之後。而有關國小學生「分數」學習的研究亦發現：國小學生對於分數值的理解十分薄弱，其對「分數」缺乏正確的量感。為進一歩瞭解國小學生對分數大小的理解及其思考策略，本研究以美國伊利諾州中部三所小學的24位四、五年級的學童為對象。藉由不同分數之大小比較及分數與整數之關係第二類測驗，進行訪問研究。針對訪問資料的分析，本研究有下列主要的發現：（1）國小學生普遍缺乏對於分數值的正確理解。學生在判定一個分數之大小時，受到其整數概念的影響。同時，學生的反應亦顯示其缺乏概念理解，而其分數估算的能力亦不足。（2）學生運用多種不同的策略以判定分數的大小。在分數與整數關係的測驗上，學生以分子或分母、分子與分母之和、操作教具或圖形、參照點、以及分子與分母的乘法關係等策略來判定一個分數之值。而在比較不同分數之大小時，學生所使用了的策略包括：只比較分子、只比較分母、分子與分母兼顧、分子與分母之差、操作教具或圖形、參照點、與比例策略點。這些策略顯示了兒童在分數上的迷失概念及其不同程度的理解。

The main purpose of this study was to investigate children’s understanding of fraction size；specifically， their ability to order fractions and to relate the size of a fraction to whole numbers. Children’s thinking strategies for tasks involving fraction─fraction comparisons and fraction─whole─number relationship were also examined.Two fraction─size tasks， which assessed children’s ability in relating the size of fractions to whole numbers（FW）and comparing the size of fraction pairs（FF）， were administered to 24 fourth─and fifth─grade children from three schools in east─central Illinois. Individual interviews were also conducted to explore children’s thinking strategies.The results revealed the following：\r1. Children’s performance on the fraction─and─whole─number─relation（FW） tasks revealed several misconceptions， including the dominance of whole─number concepts， detachment of conceptual understanding from mathematical procedures， using fraction models without knowledge of fraction size， and deficiency of estimation skill. Children’s performance on the fraction─fraction comparison tasks（FF） tasks revealed an additive type of thinking. Results also showed that children performed better on the FF tasks on the FW tasks.2. Children employed various thinking strategies on the FW and FF tasks. They used parallel strategies across two different types of tasks involving fractions. It was observed that strategies that strategies that required multiplicative thinking were related to success on the fraction─size tasks.The findings implied that instruction of fractions needs to help children recognize the differences between the concept of whole numbers and the concept of fractions， develop an understanding of the size of a fraction in relation to other numbers including whole numbers， and construct an understanding of the multiplicative relationship. Efforts need to be taken to help children build connections between the concrete representations of fractions and the idea of fractions as numbers.