Observational and experimental studies on roles of communication in mathematics learning

在信息社会的教育中，沟通能力变得越来越重要。本研究考察了初中数学课堂交往中存在的问题，并寻求改善这种状况的途径。数学能力的发展涉及到数学话语的获得：学生应该能够灵活地处理符号、公式、术语、数字、表格、图形等，以及用于交流和调查的工具。在本研究的第一年，我们在不同类型的数学课堂上进行了观察案例研究，发现了以下情况：1.即使学生在一节课上的话语频率较高，但他们的表达往往局限于短时间的反应。计算机的使用有可能将新的数学交流方式引入教室，但还没有被充分认识到。第二年，运用乔治·莱科夫的认知模型理论对中学数学教材中的数学证明话语进行了分析。美国和日本的证明教学中分别存在模块化结构和有机结构，其中“旅程”隐喻较为常见。有人认为，数学证明应被理解为体验或引导一次数学探究之旅，并对这一旅程进行说明(“讲故事”)。在此基础上，在初中进行了几何证明学习的教学实验，提出了一种在数学证明学习中加强沟通的教学方法：运用冒险故事的比喻来解决证明问题，并要求学生保留自己的笔记本，以便记录自己或他人的作品并进行反思。

The ability of communication has become increasingly important in education for the information society. This research examined problems of classroom communication in mathematics learning at junior high school, and sought a way to improve the situation. Development of mathematical ability involves acquisition of mathematical discourse : Students are expected to be able to flexibly handle symbols, formulae, terminology, figures, tables, graphs, etc.as toosl for communications as well as for investigation. In the first year of this research, ovservational case studies in various types of mathematics classroom were conducted, and the following situations were identified :1. Even if the frequency of students' utterance is rather high in a lesson, their expressions are often limited to short responses.2. The use of computer has potentials to introduce new ways of mathematical communication to the calssroom, but they have not yet been fully appreciated.3. Team-teaching has rich potentials to enhance classroom communication when communication between teachers is well organized.In the second year, mathematical discourse of proof in school textbooks was analyzed using George Lakoff's theory of cognitive model. Modular structure and organic one were found in the instruction of proof in the United States and Japan, respectively, and a "journey" metaphor was common. It was argued that mathematical proof be underestood as experiencing or guiding a journey of mathematical inquiry as well as giving accout ("telling a story") of the journey. Based on this analysis, a teaching expeirment on learning of proof in geometry was conducted in a junior high school, and a teaching method to improve communication in learning mathematical proof was developed : It uses a metaphor of adventure story for solving proof problems, and asks students to keep their own notebooks for recording their own or other's works and reflecting on them.