上海高中数学分类讨论思想及其教学研究

发布时间：2018-01-09 05:15

本文关键词：上海高中数学分类讨论思想及其教学研究　出处：《上海师范大学》2016年硕士论文　论文类型：学位论文

【摘要】：数学思想,是数学知识内容的精髓和灵魂,是对数学的本质认识.它是把数学知识的学习和能力的培养有机结合起来提高个体思维品质和数学能力的关键,更是一个人数学素养的重要内涵之一.分类讨论思想作为高中阶段最重要的数学思想之一见于教材的各个章节,在高考中具有举足轻重的地位.学生解题时分类讨论的意识不强,讨论时存在盲目性、主观性.现有研究偏重解题研究,较少教学研究.本文紧密结合教材和高考,为上海高中数学教师提供丰厚的分类讨论思想教学资料,希望能为分类讨论思想的教学提供有价值的参考.本文主要进行了如下研究:对现行上海高中数学教材做了深入研究,充分挖掘了集合、不等式、函数、三角函数、数列、平面向量、行列式、解析几何、复数、立体几何、排列组合和概率中蕴含分类讨论思想的知识点、例题和习题,对同类型题做了归类,对教材中出现的引起分类讨论的原因做了总结.然后以引起分类的原因分析了上海近六年高考试题,并对每年高考试题中分类讨论的题型、分值、讨论因素、知识载体做了统计,得出了高考常以函数、数列、解析几何、向量、集合、不等式等知识为载体考察分段函数、含参数的分类讨论、排列组合概率、分类给出的公式、设直线时考虑斜率是否存在、去绝对值讨论、不等式的运算性质、图形的不确定性、根据剩余类讨论、结论不唯一时讨论等分类讨论内容的结论.根据以上研究,结合前人研究和自身十三年的教学经验,提出了对上海高中数学分类讨论思想教学的五项教学原则和七项教学策略,并以概念课、公式推导课的教学设计为例说明如何进行分类讨论思想的教学.其中五项教学原则为渗透性原则、渐进发展原则、系统性原则、明确性原则、自我构建原则.七项教学策略:(1)充分挖掘教材中的分类讨论思想方法.给出了通过变式把一些例题、习题改编成分类讨论题的例子.(2)重视知识的发生过程,加强分类讨论思想方法的训练和培养.对概念教学不要直接给定义,对公式、定理、性质不要过早下结论.(3)搞好整理总结,进行分类讨论思想方法的概括和提炼.总结了避免或优化讨论的策略(4)加强解题指导,突破分类讨论思想的难点.(5)反复渗透,掌握分类讨论重点.对分类讨论重点考察内容,教学时要通过不同知识载体反复渗透分类讨论思想.(6)纠错练习,强化易错点.(7)重视其它数学思想方法.
[Abstract]:Mathematical thought is the essence and soul of mathematical knowledge content and the essential understanding of mathematics. It is the key to improve individual thinking quality and mathematical ability by organically combining the study of mathematical knowledge with the cultivation of ability. As one of the most important mathematical ideas in senior high school, the classified discussion thought can be found in every chapter of the textbook. In the college entrance examination has a pivotal position. Students' awareness of problem solving classification discussion is not strong, the discussion exists blindness, subjectivity. The existing research emphasis on problem solving research. This article closely combines the teaching materials and the college entrance examination to provide the rich classified discussion ideological teaching materials for the senior high school mathematics teachers in Shanghai. Hope to provide a valuable reference for the teaching of classified discussion ideas. This paper mainly carried out the following research: the current senior high school mathematics textbooks in Shanghai have done in-depth research, fully excavated the set, inequality, function. Trigonometric functions, sequence of numbers, plane vectors, determinants, analytic geometry, complex number, solid geometry, permutation and combination, and probability are the knowledge points, examples and exercises of classification and discussion. This paper summarizes the reasons that cause the classification discussion in the teaching materials, and then analyzes the questions of the college entrance examination in Shanghai in the past six years with the reasons of the classification, and discusses the types, scores and discussion factors of the questions classified in the examination questions of the college entrance examination every year. The knowledge carrier has made statistics and obtained that the college entrance examination often uses function, number series, analytic geometry, vector, set, inequality and other knowledge as the carrier to investigate the piecewise function, including the classification and discussion of parameters, the probability of arrangement and combination. The formulas given by the classification consider whether the slope exists, the absolute value is removed, the operation property of the inequality and the uncertainty of the graph are considered when the straight line is set, and discussed according to the remaining class. According to the above research, combined with the previous studies and their own 13 years of teaching experience. This paper puts forward five teaching principles and seven teaching strategies to discuss the classified teaching of high school mathematics in Shanghai. The teaching design of formula derivation course as an example shows how to carry on the teaching of the classified discussion thought, among which five teaching principles are permeability principle, progressive development principle, systematic principle and clear principle. Self-construction principle. Seven teaching strategies: 1) fully excavate the thinking method of classification discussion in the textbook. An example of adapting exercises into classified discussion questions. (2) pay attention to the process of knowledge, strengthen the training and cultivation of thinking methods of classification discussion. Do not directly define the concept teaching, and give formulas and theorems. Do not prematurely draw conclusions. 3) do a good job in sorting out and summing up ideas and methods of classification discussion. Summarize the strategies to avoid or optimize discussion. 4) strengthen the guidance of problem solving. Break through the difficulty of classification discussion. 5) repeatedly infiltrate, grasp the focus of classification discussion. In teaching, we should repeatedly infiltrate into different knowledge carriers to discuss the idea of classification. 6) the error-correcting exercises should be strengthened, and the error-prone point should be strengthened. 7) other mathematical thinking methods should be emphasized.
【学位授予单位】：上海师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：G633.6

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