“数学归纳法”单元教学设计研究

发布时间：2018-08-18 14:16

【摘要】：数学中如何进行单元教学设计,许多研究者都进行了研究,一些研究成果已开始实施于教学中。“数学归纳法”是高中数学的重要组成部分,内容体系中蕴含着深刻的数学思想方法。早期调研发现,学生只会简单机械性地应用其证明方法,并不理解其原理。基于此,笔者运用单元教学设计的思想,对“数学归纳法”进行单元教学设计,以期促进学生更好地理解“数学归纳法”的实质,做到学以致用。在查阅文献、参与教学实践的基础上,笔者认为数学单元教学设计的核心要素主要有六个:数学要素、教材要素、学情要素、标准要素、目标要素以及结构要素。基于笔者的认知水平,本文主要对教材要素、学情要素、标准要素进行了分析,其中教材要素分析了人教A版、人教B版、北师大版以及苏教版四种不同版本教科书在“数学归纳法”内容建构中呈现的内容上的特征、结构上的特征以及内容的期望,并在分析中,整合其优势,适切地进行了教学设计,为“数学归纳法”单元教学设计奠定基础;关于学情要素笔者使用问卷调查法进行分析,问卷中仿照菲施拜因和埃吉尔的研究方法对高三年级3个均已学习完“数学归纳法”的班级(实验班、重点班、普通班)进行了调查,以明了学生在学完“数学归纳法”后在其原理上理解的心理困难,从而反馈到高二学生学习“数学归纳法”时有重点、有针对性强化其理解,为教师教学设计积累经验,从而使教师在教学设计中能够聚焦学生的心理困难;标准要素主要从考试评价的角度进行分析,分析了近五年高考中涉及到“数学归纳法”的试题,利用鲍建生的难度分析法比较了其在高考试题难度方面的变化,通过高考的导向作用,笔者反思单元教学设计中例习题的处理方式,进而科学地把握“数学归纳法”单元教学设计中内容的难度及相关的题型。基于上述工作的重点,从单元目标、教学安排、逻辑分析及设计反思四个方面来完成本单元的教学设计。在单元教学设计思想的指导下,笔者利用系统的观点,环环紧扣地设计出了由四课时建构起的“数学归纳法”单元教学设计,每课时的教学设计流程包括教学目标、教学重难点、教学方法、教学过程以及教学反思。根据单元教学安排,设计的四课时分别为:数学归纳法原理的引入、规避数学归纳法证明错误、数学归纳法原理的深理解、利用数学归纳法解决问题。其设计的核心思想在于“理解”:一是理解“数学归纳法”原理,逐渐接受“数学归纳法”原理中蕴含的最重要的数学思想即递推的思想,提高学生的数学思维水平,拓展数学视野;二是理解解决问题的思路,即先归纳再猜想后证明,该思路不仅应用于解决数学问题还可应用于其他学科问题的解决,突出了解决问题的数学思想方法,并使学生了解数学结论的来源、数学家创造问题的思路,提升学生的信心以及数学素养。四个课时逐步深入,层层递进,从理解原理到应用再到理解数学思想,最终目的不仅是教会学生会做题,同时也希望能帮助教师在教学中渗透数学思想方法。
[Abstract]:Many researchers have studied how to design unit teaching in mathematics, and some research results have been put into practice in teaching. "Mathematical induction" is an important part of high school mathematics, and the content system contains profound mathematical thinking methods. Based on this, the author uses the idea of unit instructional design to carry out unit instructional design of "mathematical induction" in order to promote students to better understand the essence of "mathematical induction" and make it practical. On the basis of consulting literature and participating in teaching practice, the author thinks that the verification of mathematical unit instructional design should be carried out. Based on the author's cognitive level, this paper mainly analyzes the elements of the teaching material, including the elements of the teaching material A, B, Peking Normal University and the Soviet Edition. Textbooks in the "mathematical induction" content construction presented in the content of the characteristics, structural characteristics and content expectations, and in the analysis, integration of its advantages, appropriately carried out the teaching design, for the "mathematical induction" unit teaching design lays the foundation; on the elements of the use of questionnaire survey analysis, questionnaire By imitating the research methods of Fischbain and Egill, the middle school investigated three classes (experimental class, key class, and ordinary class) in senior three who had learned "mathematical induction" in order to understand the students'psychological difficulties in understanding the principle of "mathematical induction" after learning it, so as to feed back to senior two students to learn "mathematical induction". The standard elements are mainly analyzed from the angle of examination evaluation, and the test questions related to "mathematical induction" in the college entrance examination in the past five years are analyzed, using Bao Jiansheng's difficulty analysis method. By comparing the changes in the difficulty of the college entrance examination, the author reflects on the way to deal with the example exercises in the unit instructional design, and then scientifically grasps the difficulty of the content and the related questions in the unit instructional design of "mathematical induction". Under the guidance of the idea of unit teaching design, the author has designed the unit teaching design of "mathematical induction" which is constructed by four lessons from a systematic point of view. The process of teaching design for each lesson includes teaching objectives, teaching emphasis and difficulties, and teaching side. According to the unit teaching arrangement, the four lessons are designed as follows: the introduction of mathematical induction principle, avoiding the proof error of mathematical induction, the deep understanding of mathematical induction principle, and the use of mathematical induction to solve problems. Principle, gradually accept the most important mathematical thought contained in the principle of "mathematical induction" that is, recursive thinking, improve the level of students'mathematical thinking, expand mathematical horizons; second, to understand the thinking of solving problems, that is, after induction and conjecture, prove that this thinking can be applied not only to solve mathematical problems but also to other disciplines. Solving, highlighting the mathematical thinking and methods to solve problems, and making students understand the source of mathematical conclusions, mathematicians to create ideas to enhance students'confidence and mathematical literacy. It also hopes to help teachers infiltrate mathematical thinking and methods in teaching.
【学位授予单位】：西北师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：G633.6

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