【作者】 马文杰；

【导师】 鲍建生；

【作者基本信息】 华东师范大学 ， 学科教育， 2014， 博士

【摘要】 从学生数学学习的总体过程而言,数学学习错误,包括解题错误在某种程度上是不可避免的。因而,在数学学习过程中产生一定的数学学习错误是必然的,也是合理的。但从教学角度而言,我们又期望学生能够比较顺利地掌握相应的数学知识。因此,深入研究学生在数学学习中出现的各种错误,进行科学、合理的归因,并研究有效地避免或矫正学生数学学习错误的方法等具有重要的实践价值与理论意义。函数概念内涵丰富、思想深刻、应用广泛,是高一数学的核心知识与关键内容。另一方面,高一学生在学习函数的相应内容时,也暴露出了一系列的问题,在解决与函数有关的问题时,也出现了各种各样的错误。因此,以函数内容为载体研究高一学生的数学学习(解题)错误,具有重要的实践价值。本研究以人教版《高一数学必修1》(A版)为载体,主要研究了以下三个基本问题：(1)在解决与函数有关的问题时,高一学生主要出现哪些类型的错误?(2)导致这些解题错误的主要原因是什么?(3)如何有效地矫正高一学生的数学解题错误?在梳理与分析国内外有关学生数学学习(解题)错误的相关研究的基础上,作者确定了本研究的研究方法、分析框架和研究工具,等等。本研究用到的主要研究方法有：文献分析法、访谈法、作业(试卷)分析法、个案研究,以及问卷调查,等等,这些研究方法互相支持,互相补充,使作者在研究过程中能够不断“攻坚克难”,顺利完成研究任务。本研究构建的分析与矫正高一学生数学解题错误的基本框架为：识别解题错误、分析解题错误、矫正解题错误、评价与完善矫正方案。从一般层面分析高一学生解答与函数有关的问题的过程中出现的解题错误时,本研究主要采用以下分析框架：知识性错误、逻辑性错误、策略性错误,以及疏忽性错误。从具体层面分析高一学生在解答某一个数学问题的过程中出现的错误解答时,除了使用以上一般层面解题错误的四分类法,另外还主要采用“错误模式”和错误“复现率”对其进行分析与研究。本研究用到的基本研究工具主要有：作者专门为本研究开发的《高一学生数学学习问卷》和七套《高一数学测试卷》。通过这两个研究工具,笔者收集到了十分丰富、非常生动的第一手研究资料,为本研究的深入开展奠定了坚实的“物质基础”。在综合已有研究的基础上,作者初步构建了数学解题错误矫正的基本原则,以及数学解题错误矫正的基本框架与基本流程。并在教学实践的基础上,反思与总结了基于“解题错误”的个别辅导矫正方式和基于“解题错误”的课堂教学矫正方式。通过本研究,笔者主要得到以下结论：首先,高一学生在解答与函数有关的问题时出现的解题错误主要是知识性错误与疏忽性错误,同时,逻辑性错误与策略性错误也在解答过程中不同程度地出现。另外,通过深入分析本研究的系列测试,作者发现高一学生的数学解题错误是有一定“模式”与“结构”的。这在一定程度上可以为我们提供一个对解题错误进行分类的标准,也有利于对错因进行推断,以及合理确定矫正起点,对其进行适当矫正,等等。其次,综合已有的相关研究,并通过对本研究系列测试的分析,以及与学生的访谈、与任课老师的交流等,作者从大的方面把导致高一学生数学解题错误的主要原因归结如下：数学内容方面的原因、数学教学方面的原因,以及数学学习方面的原因。再次,个别辅导是分析错误,矫正错误的一种有效而重要的方式。个别辅导矫正比较自由、灵活,易于调整,便于深入,有利于深入观察解题者的解题过程,有利于发现其个别化的错因。通过个别辅导,可以对学生的解题错误理解的更深入,更全面。另外,通过个别辅导矫正,可以和学生进行“深度交流”,可以了解学生的个性特点、习惯爱好、思想动向,等等。这都对研究与矫正学生的数学解题错误有一定益处。第四,基于“解题错误”的课堂教学矫正方式完全有潜力发展成为一个高效的错误矫正方式。基于“解题错误”的课堂教学矫正的取材十分方便,操作简单易行。基于“解题错误”的课堂教学矫正的立足点是学生的“解题错误”,基本的教学素材也是学生的“解题错误”,以及学生在教学过程中即时生成的一些教学资源,基于“解题错误”的课堂教学矫正的最终目的,则是为了更好地矫正学生的解题错误,最大可能地消除学生的错误认识。更多还原

【Abstract】 In the overall processes of learning mathematics, students’mathematics learning errors, including errors in solving mathematical problems are inevitable to some extent. But from teacher’s perspective, we expect students to be able to grasp mathematical knowledge more successfully. Therefore, in-depth study of students’all kinds of errors in learning mathematics is very important. Error analysis and error correction are very important for mathematics instruction and mathematics education research.The concept of function is rich, profound, and widely used. It is one of the core topics in high school level mathematics. However, senior high school students are confronted a series of problems and have showed variety of errors in solving problems related to the topic of function. Therefore, an empirical study of problem solving errors in function learning by10th graders is needed, which has important theoretical and practical value.The study is based on high school mathematics textbook (compulsory one A version by PEP), and the three focuses are:(1) what types of errors are committed when senior one students solving mathematical problems related to function?(2) what are the main causes of these errors?(3) how could students correct these errors effectively?By reviewing researches on errors in mathematics learning (mathematical problem solving), the author selected appropriate research methods, elaborated the analytical frame work and chose the research tools, and so on. The main research methods used in this study are:literature analysis; interviews; task(paper) analysis; case studies and surveys, etc. These methods support and complement each other, so that we can successfully" overcome difficulties", and successfully complete the research tasks.The study analyzed students’errors in mathematics problem solving from the following perspectives:identifying problem-solving errors, analyzing error, correcting error, evaluating and improving correction program, which form the basic framework of the study. When analyzing errors students made in solving function-related questions, a specific analytical framework were used from four categories of errors: mathematical error, logic errors, strategic errors and careless errors. In addition,"error model" and error "reproduction rate" were also used in analyzing errors made by senior one students in this study. The textbook used by student participants is the compulsory one A version by PEP, which is the main teaching material analyzed in this study.The main research tools used in this study are:"Questionnaire on senior one students learning mathematics", and seven sets of "senior one math tests", which were specifically developed by the author. By the two research tools, a set of rich, vivid first-hand research data were collected, formed a solid "material base" for the research.Based on the existing researches, the author initially built the basic principles, the basic framework for error correcting, and the basic processes of correcting mathematical problem solving errors. The summary and reflection of individual counseling and classroom correction based on "problem-solving errors" were collected from the teaching practice.Through this study, the author has obtained the following conclusions:Firstly, more mathematical errors and careless errors have appeared in solving mathematical problems, rather than logic errors and strategic errors in addition, through in-depth analysis of series of the tests, it has been found that errors made by the10th graders have some "models" and "structures", which contributes to analyzing causes of errors, clarifying errors, and correcting errors.Secondly, based on the existing researches, the analysis of the seven tests in the study, as well as interviews with the students and the teachers, the author has drawn that the main causes of students’mathematical problem solving errors are: mathematics content factors, mathematics teaching factors and mathematics learning factors.Thirdly, individual counseling is an effective and important method for analyzing and correcting errors. Individual counseling is flexible, easy to be adjusted. Through individual counseling, the process of problem solving can be understood more deeply and more comprehensively. In addition, through individual counseling, we can understand the characteristics, habits, hobbies, ideological trends of the student’s, and individual counseling can also promote feelings between teachers and students, which all are benefit to study and correct students’errors.Fourthly, classroom correction based on "individual problem-solving errors" seems to be an efficient error correction method, as it is very convenient, and easy to be operated. Mathematics teaching, math homework, and math tests are important sources of collecting students’ mathematical problem solving errors. Modern technological tools can provide great convenience for collecting students’solving errors. Basic teaching material of classroom correction based on "individual problem-solving errors" is students’ problem-solving errors, All instructing activities focus on students’ problem-solving errors, the ultimate goal of which is to correct students’ errors more effectively and to eliminate students’ misconceptions more thoroughly更多还原