家庭作业里的教学与课程潜力:对几何证明作业的讲解与点评——上海的案例研究

Pedagogical and Curriculum Potentials of Homework:Expalinging and Commenting on Homework in Geometric Proofs——A Case Study of Shanghai

作为一项案例研究,文章聚焦于上海东南部的一所普通初级中学,探讨一位有经验的教师(王老师)如何在课前讲解与点评学生的几何证明作业。研究发现体现在三个主要方面:第一,通过话语分析揭示这位教师讲解和点评作业惯常的行为结构,她通过IRE/F(教师用提问开启-学生回答-教师评价/反馈)课堂话语引导学生共同参与讲解,点评学生的错误及其原因,帮助他们回忆、再联结和再建构他们先前所学的知识,通过这样的即时反馈帮助学生在学习新知识前及时纠正错误、强化巩固。通过数个小时的时间批改学生的几何作业,分析学生的思维并对个别学生进行单独辅导之后,王老师的讲解在呈现出细节化、层次化和结构化的同时从学生的错误中抓住数学学科的本质。第二,通过分析教材和批改过的学生作业中的错误,发现学生学习过程中出现的错误和困难并不是新学的或难以理解的定理,而是缺乏几何证明书写过程中所需要遵循的基本规范与逻辑思维习惯。第三,题目的变式设计的传统营造了精心设计的多样化的问题背景,使得教师通过批改、辅导、讲解和点评作业能够及时发现并解决学生的学习问题,进而帮助学生理解和掌握基本的演绎推理技能和习惯。借助家庭作业,本研究揭示了上海地区常见的数学教学实践背后潜藏的维度,以帮助我们重新思考家庭作业在数学学科的教与学中的作用与潜力。

This case study focuses on how an experienced teacher in a regular junior secondary school in the Southwest of Shanghai explained and commented on student homework at the beginning of her lessons on geometric proofs. Three major findings are reported: First, a discourse analysis reveals a structured set of her routine instructional actions that she executed in involving the students in an IRE/F-patterned instructional discourse to get students to recall, reconnect and reconstruct their earlier learning in ways to allow them to make timely corrections. Hours of marking student homework, ‘analyzing' student thinking and tutoring individual students made the explaining detailed, multifaceted, structured and targeted at the mathematical substance embedded in the errors. Second, analysis of the curriculum materials and marked student work vis-à-vis the errors explained finds that the errors or difficulties of learning were not with the new or difficult theorems but with the fundamental norms to follow and habit of thinking required in writing geometric proofs. Third, the tradition of design with variation was found to have students discern and master the fundamental deductive reasoning skills and habit under diverse, carefully designed problem settings while allowing the teacher to pick up their learning issues and address them timely through marking, tutoring and explaining and commenting on student homework. These uncover the hidden dimensions of a common practice of mathematics teaching in Shanghai mediated through teachers' homework practice and help us rethink the role and potentials of homework in mathematics teaching and learning.