分数度量意义发展的认知根基及轨迹:分数图式进阶理论

The Cognitive Root and Trajectory of Measuring the Meaning of Fraction:The Conceptual Progression of Fractional Schemes

斯特芬(Steffe)、撒冷(Tzur)等人通过长期的质的建构主义教学实验研究,对西方儿童的分数学习轨迹进行了探索,提出的分数图式进阶理论为学生对分数度量意义理解的发展提供了认知依据.撒冷的"活动-效果关系"反省理论可以解释图式的构建和转化机制,迭代和均分是分数图式构建的两种重要认知操作,而分数图式进阶模型共包括8个进阶水平,前4个图式主要基于迭代操作,后4个图式主要基于均分操作.总之,分数概念的本质不是"部分-整体",而是度量意义;学生的分数图式是对整数计数图式的顺应;迭代和均分操作能够促进学生对分数度量意义的理解.

Through longitudinal and constructivism-based qualitative teaching experiments,Steffe,Tzur,and other researchers explored western children’s learning trajectories of measurement of fractions and proposed the Conceptual Progression of Fractional Schemes,which is the cognitive root of children’s development in the meaning of fraction.The theory of“reflection on the activity-effect relationship”proposed by Tzur could explain the construct and transformation of schemes,and iterating and partitioning are two types of cognitive operations underlying the construction of fractional schemes.The Conceptual Progression Model of Fraction Schemes includes 8 levels;the first 4 levels are based on iterating and the last 4 schemes on partitioning.We argue that a fraction is not“part of whole”but rather a multiplicative relation with a meaning of measurement.Students’fractional reasoning is a reorganization of their whole number reasoning;the operation of iterating and partitioning could improve students’understanding of fractions as multiplicative relations.