Arithmetic Project: Fractional Schemes

这个项目的目的是进行一项纵向研究，在两年的时间里，调查儿童在解决涉及分数的问题情景时产生的建设性路径，以及他们在努力解释自己的解决方案时形成的沟通模式。12名儿童将在三年级和四年级时参加这项研究。研究人员将开发儿童生成分数的建构过程的模型，还将开发学生与学生和学生与教师社会互动的交流模式的模型。几个研究问题构成了这项纵向研究的基础：儿童构建和操作不同复合单元的能力在构建儿童可以利用整体中部分的多样性进行分数量化的部分-整体关系中至关重要吗？传统的分数符号化是否需要分数概念的先验发展建构过程，才能使其对儿童产生意义？孩子们会生成他们自己的分数运算算法吗？这些算法与传统算法相似吗？儿童在认知上从社会互动中获得了什么？在什么情况下？社会互动的哪些方面有助于儿童分数概念的发展？根据孩子的年龄不同，社交互动的作用是否有所不同？本课题的研究结果将有助于理解1)儿童长期(两年)对分数的概念化，2)儿童自然数知识对分数构造性发展的影响；3)社会互动(与同伴和老师)在儿童分数概念的构建和演变中的作用；4)将用自然语言在心理上和言语上形成的概念符号化所需的认知需求。

The purpose of this project is to conduct a longitudinal study that will investigate, over a period of two years, teh constructive paths that children generate when solving problematic situations that deal with fractional numbers, and the communication patterns that they develop in their efforts to explain their solutions. Twelve children will participate in the study when they are in third grade and again when they are in fourth grade. The researchers will develop models of the children's constructive processes in generating fractional numbers and will also develop models of the communication patterns of student-student and student-teacher social interaction. Several research questions form the basis of this longitudinal study: Is children's ability to construct and operate with different composite units essential in the construction of part-whole relations that children can quantify fractionally using the multiplicity of the parts in the whole? Does conventional symbolization of fractions require a priori developmental constructive processes of the concept of fractions before it becomes meaningful to a child? Do children generate their own algorithms to operate with fractions? Are these algorithms similar to the conventional ones? What do children gain cognitively from social interaction and under what circumstances? What aspects of social interaction contribute to children's advances of their fraction concepts? Are there differences in the role of social interaction depending on the child's age? The results of this project will contribute to the understanding of 1) children's conceptualizations of fractions over a long period of time (two years); 2) the influence of their knowledge of natural numbers on the constructive development of fractional numbers; 3) the role of social interaction (with peers and teacher) in the construction and evolution of children's concepts of fractions; 4) the cognitive demands required to symbolize concepts that had been developed mentally and verbalized using natural language.