高中生向量理解水平的调查研究

发布时间：2019-01-07 21:08

【摘要】：20世纪以来,数学理解(Mathematical understanding)问题,业已成为当今数学教育研究的又一个热点话题。随着课程改革的不断推进,许多国家都已将向量作为高中数学的教学内容。我国在1996年颁发的《全日制普通高级中学数学教学大纲》将“平面向量”列为了必修内容,将空间向量列为处理空问几何问题的一种方法,从此向量正式进入我国的高中数学课程；2003年教育部颁布的《普通高中数学课程标准(实验)》中在必修模块4和选修系列2-1当中分别设置了平面向量和空间向量。现在向量已经成为高中数学的重要内容,现实教学中高中生对向量的理解现状如何?教师如何根据处于不同理解层次学生的情况设计和实施教学,选择教学策略成了高中教师关注的问题。本文以APOS理论为基础,以向量概念为载体,通过测试调查和访谈法对扬州市289名高中生和3位数学教师进行了“高中生向量理解水平的调查研究”,具体的研究问题包括：(1)调查向量单元学习后高中生的理解现状如何；(2)探讨分析学生达到某种理解水平的原因是什么；(3)基于学生的理解水平,有效的数学教与学的策略是什么。笔者对调查得来的数据进行了定量和定性分析,得到了以下结论： (1)关于向量操作阶段的理解,根据调查统计发现,高一学生有67.53%是优秀,调查的学生当中都在合格及以上；高三年级学生有38.21%达到了优秀,另有45.75%的学生处于良好水平。两个年级的学生已基本能达到向量操作阶段的常规水平。学生基本能判断一个量是否为向量,能理解物理矢量与标量之间的本质区别。 (2)学生对各种类型的向量的特征理解存在差别。在考察的过程,学生往往容易忽略对零向量的考虑,缺乏分类讨论的思维习惯。 (3)学生对向量各种表征形式之间的转化的理解存在一定的单向性。从图像形式转为符号和坐标形式、符号与坐标之间的转化较为容易,但是从符号形式转化为图像形式解决问题较困难。 (4)高三学生比高一学生在运用向量解决问题时有较高的灵活性。但从知识网络联系看,两个年级的网络内容都不够丰富,并没有呈现出随年级的线性增长,联系的对象以下位概念为主,缺乏对平行概念的联系。结合本研究的结果,笔者提出了几点教学要求：(1)丰富感性经验,从直观上理解概念；(2)从多种表征方式理解概念；(3)注意加强知识点之间的联系。并对师范生的培养提出两点建议：(1)随着专业知识的积累,注意初等数学中相关概念的理解；(2)依据师范性开展研究性学习,自觉提高对初等数学知识的理解水平。
[Abstract]:Since the 20th century, mathematics understanding of (Mathematical understanding) has become another hot topic in mathematics education. With the development of curriculum reform, many countries have taken vector as the teaching content of high school mathematics. In 1996, the Mathematics syllabus of Full-time General High School in our country made "plane vector" a compulsory subject, and space vector as a method to deal with the problem of space geometry. Since then vector formally entered our senior high school mathematics curriculum; In the "ordinary Senior Middle School Mathematics Curriculum Standard (experiment)" issued by the Ministry of Education in 2003, plane vector and space vector were set in compulsory module 4 and elective series 2-1, respectively. Now vector has become an important part of senior high school mathematics. What is the current situation of high school students' understanding of vector in practical teaching? How to design and implement the teaching according to the situation of students at different levels of understanding and how to choose teaching strategies have become the problem that high school teachers pay attention to. Based on the theory of APOS and the concept of vector, this paper makes an investigation on the level of vector understanding among 289 senior high school students and 3 mathematics teachers in Yangzhou by means of test investigation and interview. The specific research questions include: (1) the current situation of high school students' understanding after studying vector unit; (2) explore and analyze the reasons for students to reach a certain level of understanding, (3) based on the level of understanding of students, what are the effective strategies of mathematics teaching and learning. The author makes a quantitative and qualitative analysis of the data obtained from the investigation, and draws the following conclusions: (1) the understanding of the stage of vector operation, according to the survey statistics, shows that 67.53% of the first year students are excellent. All the students surveyed were qualified and above; 38.21% of the students in Grade 3 were excellent, and 45.75% of the students were in good level. Students in two grades have basically reached the normal level of vector operation. Students can basically judge whether a quantity is a vector and understand the essential difference between a physical vector and a scalar. (2) there are differences in students' understanding of the characteristics of various types of vectors. In the course of investigation, students tend to ignore the consideration of zero vectors and lack the habit of thinking about classification and discussion. (3) the students' understanding of the transformation between vector representations is unidirectional. From image form to symbol form and coordinate form, the transformation between symbol and coordinate is easy, but it is difficult to solve the problem from symbol form to image form. (4) Senior three students have higher flexibility in using vector to solve problems. But from the point of view of knowledge network connection, the network content of the two grades is not rich enough, and does not show the linear growth with grade, the object of connection is mainly the concept of the following position, and lacks the connection to the concept of parallelism. Combined with the results of this study, the author puts forward several teaching requirements: (1) enriching perceptual experience, understanding concepts intuitively; (2) understanding concepts from various representations; (3) paying attention to strengthening the relationship between knowledge points. Two suggestions are put forward for the cultivation of normal school students: (1) with the accumulation of professional knowledge, attention should be paid to the understanding of the related concepts in elementary mathematics; (2) the research study should be carried out according to the teachers' nature, and the level of understanding of elementary mathematics knowledge should be improved consciously.
【学位授予单位】：扬州大学
【学位级别】：硕士
【学位授予年份】：2012
【分类号】：G633.6

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