Process of change of teaching practice and children's proportional reasoning by making aware of a knowledge acquisition model

本研究通过与教师合作准备、实施、反思和修改基于知识获取模式的比率和比例课程，为教师提供了思考教材中术语、符号、计算和/或公式的教学与发展学生比例推理之间的关系的机会。本研究的目的是探讨教师在教学中面对数学模型认知挑战时的思维过程。2003年，我们基于“小学生内化和创造关于比例和比例的数学符号”的模型，合作开发了“单位数量”、“小数乘以小数”和“小数除法”三个内容单元的教案。在老师在课堂上实施教案的那段时间里，包括我在内的几个人观课并学习了…一些注意力集中的小学生出现更多的暗示性过程和变化。对这些小学生的观察结果尽早通知了老师。每节课结束后，老师都会对自己的教学进行反思，并就学生的思维、笔记等注意事项做一个简短的报告。我们大约每周有一次讨论的时间。2005年，对数据进行了汇总，并对教师和学生进行了分析。在教师方面，共同创立的“课堂上小学生的转化”视角为研究者和教师提供了一种讨论的情境。同时，研究人员提出的让学生画图的建议引起了老师的注意，有助于深化我们之间的讨论。这种教学行为创造了一种情景，在这种情景中，我们可以了解学生学习数学的概念，在这种情景中，我们可以更具体地谈论学生的转变。创设了教师的数学教学观得以体现和体现的情境。对于小学生来说，教师让小学生画图形的尝试促进了小学生将图形内化为思维工具和学习数学的规范的过程。本研究对弥合模型这一研究成果与实际教学实践之间的差距的一个启示是，教学工具的重要性，有助于在教学中具体化模型，并在讨论中提出教师的数学教学理念。在这项研究中，让小学生画图形就是这样一种工具。较少

In this study, through a collaborative effort with a teacher in preparing, implementing, reflecting and revising lessons on ratio and proportion based on a knowledge acquisition model, opportunities were provided for the teacher with thinking about the relationship between teaching of terms, notations, calculations and/or formulas in the textbook on the one hand, and developing pupils' proportional reasoning on the other hand. The purpose of the study is to investigate the thinking process of the teacher when he faced a challenge in making aware of the model in his teaching.In 2003, based on a model of "pupils' internalization and creation of mathematical notations concerning ratio and proportion," we developed lesson plans on three content units : "quantity per unit," "multiplication by decimals," and "division by decimals" collaboratively. During the time that the teacher implemented the lesson plans in the classroom, several people including myself observed the lessons and studied t … More hinking processes and changes of some focused pupils intensively. Results of observation of these pupils were informed to the teacher as early as possible. After every lesson, the teacher reflected on his teaching and made a brief report about notices of pupils' thinking and notations and so on. We also had time for discussion about once in a week.In 2005, the data were summarized and analyses of the teacher and the pupils were made. With respect to the teacher, a jointly-created perspective "transformation of pupils in the classroom" provided a situation of discussion between the researcher and the teacher. At the same time, a proposal of letting the pupils draw figures by the researcher attracted the teacher's attention and contributed to deepening the discussion between us. This act of teaching produced situations in which we can learn about pupils' conceptions of learning mathematics and situations in which we can talk about pupils' transformations more concretely. It also produced situations in which the teacher's conception of mathematics teaching is manifested and reflected. With respect to the pupils, the teacher's attempts of letting the pupils draw figures enhanced their process of internalizing the figures as a tool of thinking and their norms of learning mathematics.One implication of this study for bridging gaps between the model, an outcome of research, and actual teaching practice, is the importance of pedagogical tools that serve to concretize the model in the teaching and bringing out teacher's conceptions for mathematics teaching in the discussion. In this study, to let pupils draw figures was such a tool. Less