EXPRESSION FRAME OF THINKING ACTIVITY FOR LEARNING MATHEMATICS

我们研究的主要结果如下：(1)和(2)(1)在普通的数学课上，大多数老师都试图把课本上写的数学概念或知识灌输给学生。在这项研究中，我们开发了一种新的教学过程，所有学生都可以利用以下(a)和(b)定义的“表达框架”，自己构建教科书中所写的数学概念和知识。(a)学生可以利用“表达框架”构建思维活动的叙事。(b)任何“表达框架”都不代表任何思维活动的叙述。(2)我们可以认为“三个我”m1、M2和MS分别对应以下活动(1)、(2)和(3)。(1)与可见的注意对象对话。(2)在脑海中与注意对象对话。(3)结合(1)和(2)构建一个新的对话。许多数学教育研究者都强调数学课是学生通过自己的努力形成数学概念和知识的地方。但没有一种方法能让学生自己建构数学概念和知识。在本研究中，我们建立了一个满足以下条件的教与学过程原则。(A)学生脑海中的“三个我”Ml， M2和M3能够构建教科书中所写的数学概念或知识。(B)学生能够利用“表达框架”将自己的思维与其他同学的思维结合起来。

Main results of our research are given by the following (1) and (2)(1) In ordinary math classes, most of the teachers try to infuse mathematical concepts or knowledge written in the textbook to the students. They then encounter sometimes with serious difficulties to make students to form clear images of notions in their minds, We have developed , in this research, a new teaching-learning process where all students can construct the mathematical concepts and knowledge written in the textbook by themselves by making use of "Expression Frame" defined by the following (a) and (b) .(a)Students may construct a narrative of thinking activities by making use of an of "Expression Frame".(b)Any "Expression Frame" does not represent any narrative of thinking activities.(2) We may consider the" Three myself' M 1, M2 and MS corresponding to the following activities (1), (2) and (3) respectively. (1) Dialogue with visible object of attention. (2)Dialogue with object of attention in mind.(3) Construct a new dialogue by combining (1) and (2). Many researchers on mathematics education have emphasized that math class where the students can make Mathematical concepts and knowledge by their own efforts. But none of them shows a method by which the students can construct the mathematical concepts and knowledge by themselves. In this research, we have established a principle of teaching-learning process which satisfies the following.(A)" Three myself' Ml, M2 and M3 in students' mind are able to construct mathematical concept or knowledge written in the textbook.(B)A student is able to combine his/her thinking with other student's thinking by making use of the "Expression Frame".