三角函数教材的解读与重构
发布时间：2019-02-21 17:58
【摘要】：三角学是在解决天文学、物理学实际问题过程中而产生与发展起来的,对数学与自然科学产生了深远的影响.中学阶段主要包括三角函数、正弦定理与余弦定理等内容,三角函数蕴含着深刻的数学思想,研究三角函数的教材对教材编写和三角函数的教学实践与教学理论均有重要意义.本文以北京师范大学出版社、江苏科学技术出版社、人民教育出版社出版的教材为参考,以初中锐角三角函数、高中三角函数为研究对象.重点研究三角函数概念的产生与发展.本研究以质性研究方法为主.通过消化吸收已有的数学史知识、数学教学理论知识、数学学习心理知识、数学方法论知识等,结合数学教学的课堂观察与研课,对教材内容作出解读,在此基础上对教学内容进行重构.研究成果主要有:●初中部分三个版本教材为形成锐角三角函数概念都创设了问题情境,北师版教材从梯子陡坡程度、苏科版教材从台阶的坡度、人教版教材从绿化荒山的角度创设问题情境,这些问题并没有真正揭示出锐角三角形相似比的不变性这一深刻的数学思想,事实上,将之作为相似三角形的问题情境更合适.如何通过相似三角形引导学生发现直角三角形的相似不变量是问题的关键.●高中部分三个版本教材在任意角、弧度制、任意角的三角函数、三角函数的诱导公式等内容的编写风格不同.北师版教材直接给出相关知识.在弧度制、任意角的三角函数、三角函数的诱导公式设计上均以单位圆为载体,教材希望以单位圆为载体给出上述知识,但是知识间的关系不清晰.同时,有些内容的编写出现疏漏.没有体现三角函数的科学价值与应用价值,也没能体现出三角函数蕴含的数学思想.苏科版教材通过问题情境引发概念的生成,并以引言中的相关问题统领知识间的关系,知识之间的关系清晰.但在编写方面也存在一些瑕疵,例如,在静态直角三角形中标记动态的角的旋转符号等.三角函数缘何产生以及它真正的科学价值没有在教材中体现出来.表现出的主要是这些知识之间的数学关系,但是有些关系比较牵强.人教版教材在三角函数章引言上给出了宇宙天体运行图,但是在各节中却没有运用这些有价值的天文学背景为三角函数相关知识的形成提供基础.人教版教材在与读者互动方面设计较好,但是各节的问题之间缺乏联系,且有些是无效问题.三角函数揭示的天文学或物理学中旋转运动与直线运动间的关系等很少在教材中体现.本文以真实且历史上具有重要影响的问题情境——摆线统领任意角、弧度制、任意角的三角函数、三角函数的诱导公式.这一问题情境既是学生生活中常见的现象又满足学生的数学现实.以启发性问题引领学生解决物理问题的同时关注从数学内部建构数学知识等,引导学生在解决真实的物理问题过程中建构三角函数知识,并揭示三角函数形成的根源及其应用价值.
[Abstract]:Trigonometry is developed and developed in the process of solving the practical problems of astronomy and physics, and has a far-reaching influence on the mathematics and the natural science. The middle school stage mainly includes the trigonometric function, the sine theorem and the cosine law and so on, the trigonometric function contains the deep mathematics thought, the teaching material of the study of the trigonometric function is of great significance to the teaching practice and the teaching theory of the preparation of the teaching material and the trigonometric function. This paper is based on the teaching materials published by the Beijing Normal University Press, the Jiangsu Science and Technology Press, the People's Education Press, and the high school triangle function and the high school triangle function as the research object. This paper focuses on the generation and development of the concept of the trigonometric function. This study is based on the qualitative research method. Through the digestion and absorption of the existing knowledge of the mathematical history, the theoretical knowledge of the mathematical teaching, the psychological knowledge of mathematics learning, the knowledge of the mathematical methodology, etc., the content of the teaching materials is interpreted in combination with the classroom observation and the research of the mathematics teaching, and the content of the teaching is reconstructed. The research results are as follows: the three editions of the teaching materials in the junior middle school form the problem context for forming the concept of the sharp-angle triangle function, the teaching material of the north division is from the slope of the ladder, the teaching material of the Su-ke edition is changed from the slope of the step, the teaching material of the human version creates the problem context from the angle of the green barren hill, These problems do not really reveal the invariance of a sharp-angle triangle-like ratio, which is a profound mathematical thought. In fact, it is more appropriate to use it as a similar triangle. How to guide students through similar triangles is the key to the problem. The writing style of the three versions of the textbook in the high school is different from the writing style of any angle, the radian system, the trigonometric function of any angle, the induction formula of the trigonometric function, and the like. The North Division teaching material directly gives the relevant knowledge. In the radian system, the trigonometric function of any angle and the induction formula of the trigonometric function are designed with the unit circle as the carrier, and the teaching material hopes to give the above-mentioned knowledge in the unit circle as the carrier, but the relation between the knowledge is not clear. At the same time, there is an omission in the preparation of some of the content. The scientific value and application value of the trigonometric function are not reflected, and the mathematical idea contained in the trigonometric function is not reflected. The teaching material of the Su Ke version is the generation of the concept through the situation of the problem, and the relationship between the knowledge and the knowledge is the clear relationship between the knowledge and the knowledge in the introduction. however, there are also some blemish in that preparation of, for example, the rotation symbol of the dynamic angle in a static right-angled triangle, and the like. The origin of the trigonometric function and its real scientific value are not reflected in the teaching material. The main thing to show is the mathematical relationship between these knowledge, but some of them are far-fetched. In the introduction of the chapter of the trigonometric function, the teaching material of human teaching has given the operation diagram of the universe, but it is not used in each section to provide the basis for the formation of the related knowledge of the trigonometric function. The teaching material of the teaching version is well designed in the interaction with the reader, but there is a lack of contact between the problems of each section, and some are invalid. The relation between the rotational movement and the linear motion in the astronomy or physics revealed by the trigonometric function is seldom reflected in the teaching material. In this paper, the induction formula of any angle, radian system, trigonometric function of arbitrary angle, and trigonometric function is given in the real and historical context of the problem with the important influence. This problem is not only the common phenomenon in the life of the students, but also the students' mathematical reality. In order to lead the students to solve the physical problems with the heuristic problems, the paper focuses on the construction of the mathematical knowledge from the interior of the mathematics, and guides the students to construct the knowledge of the trigonometric function in the process of solving the real physical problems, and reveals the root causes of the formation of the trigonometric function and its application value.
【学位授予单位】：广州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：G633.6
，
本文编号：2427751
[Abstract]:Trigonometry is developed and developed in the process of solving the practical problems of astronomy and physics, and has a far-reaching influence on the mathematics and the natural science. The middle school stage mainly includes the trigonometric function, the sine theorem and the cosine law and so on, the trigonometric function contains the deep mathematics thought, the teaching material of the study of the trigonometric function is of great significance to the teaching practice and the teaching theory of the preparation of the teaching material and the trigonometric function. This paper is based on the teaching materials published by the Beijing Normal University Press, the Jiangsu Science and Technology Press, the People's Education Press, and the high school triangle function and the high school triangle function as the research object. This paper focuses on the generation and development of the concept of the trigonometric function. This study is based on the qualitative research method. Through the digestion and absorption of the existing knowledge of the mathematical history, the theoretical knowledge of the mathematical teaching, the psychological knowledge of mathematics learning, the knowledge of the mathematical methodology, etc., the content of the teaching materials is interpreted in combination with the classroom observation and the research of the mathematics teaching, and the content of the teaching is reconstructed. The research results are as follows: the three editions of the teaching materials in the junior middle school form the problem context for forming the concept of the sharp-angle triangle function, the teaching material of the north division is from the slope of the ladder, the teaching material of the Su-ke edition is changed from the slope of the step, the teaching material of the human version creates the problem context from the angle of the green barren hill, These problems do not really reveal the invariance of a sharp-angle triangle-like ratio, which is a profound mathematical thought. In fact, it is more appropriate to use it as a similar triangle. How to guide students through similar triangles is the key to the problem. The writing style of the three versions of the textbook in the high school is different from the writing style of any angle, the radian system, the trigonometric function of any angle, the induction formula of the trigonometric function, and the like. The North Division teaching material directly gives the relevant knowledge. In the radian system, the trigonometric function of any angle and the induction formula of the trigonometric function are designed with the unit circle as the carrier, and the teaching material hopes to give the above-mentioned knowledge in the unit circle as the carrier, but the relation between the knowledge is not clear. At the same time, there is an omission in the preparation of some of the content. The scientific value and application value of the trigonometric function are not reflected, and the mathematical idea contained in the trigonometric function is not reflected. The teaching material of the Su Ke version is the generation of the concept through the situation of the problem, and the relationship between the knowledge and the knowledge is the clear relationship between the knowledge and the knowledge in the introduction. however, there are also some blemish in that preparation of, for example, the rotation symbol of the dynamic angle in a static right-angled triangle, and the like. The origin of the trigonometric function and its real scientific value are not reflected in the teaching material. The main thing to show is the mathematical relationship between these knowledge, but some of them are far-fetched. In the introduction of the chapter of the trigonometric function, the teaching material of human teaching has given the operation diagram of the universe, but it is not used in each section to provide the basis for the formation of the related knowledge of the trigonometric function. The teaching material of the teaching version is well designed in the interaction with the reader, but there is a lack of contact between the problems of each section, and some are invalid. The relation between the rotational movement and the linear motion in the astronomy or physics revealed by the trigonometric function is seldom reflected in the teaching material. In this paper, the induction formula of any angle, radian system, trigonometric function of arbitrary angle, and trigonometric function is given in the real and historical context of the problem with the important influence. This problem is not only the common phenomenon in the life of the students, but also the students' mathematical reality. In order to lead the students to solve the physical problems with the heuristic problems, the paper focuses on the construction of the mathematical knowledge from the interior of the mathematics, and guides the students to construct the knowledge of the trigonometric function in the process of solving the real physical problems, and reveals the root causes of the formation of the trigonometric function and its application value.
【学位授予单位】：广州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：G633.6
，
本文编号：2427751
本文链接：https://www.wllwen.com/shoufeilunwen/sklbs/2427751.html
