Cognitive Effects of Multiple Representations of Mathe- matical Concepts: An Exploratory Study

数学学习不仅仅是简单地记忆规则和程序。不幸的是，目前学习的大部分数学都是这种类型的。然而，一种共识正在形成，即学生理解重要数学概念的必要条件是他们拥有这些概念的多个表示系统，并能够在它们之间顺利有效地转换。这个项目将研究我们如何有效地为学生的多种表征系统的发展进行教学，以及这种教学如何影响他们的概念和技能的发展。本项目将设计两个计算机程序；一个用于十进制数，一个用于代数。这两个项目将同时呈现“关联”的代表性系统。学生将通过直接在一个代表系统上行动来解决问题，他们的行动将自动反映在另一个代表系统上。七年级的一个班将用五周的时间学习小数运算。表达式、恒等式、线性方程和求解线性方程的概念将在一学年的时间里与初等代数八年级学生一起学习。基本研究方法将是建构主义教学实验的研究方法。通过精心策划的个人和小组访谈协议，将构建一个（多个）学生知识模型，该模型描述了所教授的表征之间的关系。此外，亦会发展有助理解数学的课程材料。拟议的研究具有挑战性，并将导致数学和科学教育的重大进展。首席研究员在数学教育界受到高度重视，这反映在这个精心设计的研究项目中。

Mathematics learning consists of more than simply memorizing rules and procedures. Unfortunately, much of the mathematics currently being learned is of this type. A concensus is emerging, however, that a necessary condition for students to understand important mathematical concepts is that they posess multiple representational systems for those concepts and be able to translate smoothly and efficiently among them. This project will study how we can teach effectively for students' development of multiple representational systems and how such instruction affects their development of concepts and skills. Two computer programs will be designed for this project; one for decimal numeration and one for algebra. Both programs will present simultaneous, "linked" representational systems. Students will solve problems by acting directly upon one representational system with their actions being automatically reflected in the other. Operations on decimals numbers will be studied for five weeks with a class of seventh graders. Concepts of expression, identity, linear equation, and solving linear equations will be studied with pre-algebra eighth graders over a school year. The basic research methodology will be that of the constructivist teaching experiment. Through carefully planned individual and small group interview protocols, a model (models) of student knowledge will be constructed which describes the relationship beween the representations taught. Curriculum materials to facilitate mathematical understanding will also be developed. The proposed research is challenging and should lead to significant advances in mathematics and science education. The Principal Investigator is highly regarded in the mathematics education community and this is reflected in this well conceived research project.