Function and position of vectors

The emphasis on "vector" knowledge is one of the important contents of this high school textbook reform. Then, in the process of compiling new mathematics textbooks, how to understand the "vector" content under the guidance of new curriculum standards? What practical significance and far-reaching influence will it have on the whole senior high school mathematics education to add "vector" content to senior high school mathematics textbooks? What other chapters related to "vector" need to be paid attention to and improved when teaching with new textbooks? Thinking about these problems made me investigate the current situation of vector knowledge teaching. Vector knowledge has a very important position and educational value in middle school, and its instrumental characteristics are reflected in many branches of mathematics, especially in advanced mathematics and analytic geometry, the idea of vector permeates very widely! However, as a compulsory course in middle school, teachers and students pay more attention to plane vector than space vector, and the advantages of space vector in solving solid geometry are irreplaceable by traditional knowledge and methods. More importantly, it is of great benefit to cultivate students' mathematical ability and literacy, which needs the full attention of front-line teachers! Through the situation reflected in the questionnaire and the interviews with front-line teachers in the process of questionnaire distribution and collection, the author understands that a considerable number of front-line teachers are evasive about space vectors, which is very unfavorable for the implementation and promotion of the new curriculum! From the questionnaire, we can see that teachers still rely heavily on traditional methods. When dealing with the relationship between vector method and traditional method, they often focus on traditional method. Even if they are not very skilled in using them, they should be compared with traditional methods. This result will often lead to the shortage of class time, students are easily confused and bring unnecessary extra burden, so teachers will have the illusion that they are still good! Some teachers have realized the important educational value of vector knowledge, but due to the stylization and fixed mode of the original knowledge, especially the old teachers, it is urgent to solve the training of the new curriculum, which can supplement the lack of knowledge in time and make full preparations for the promotion and implementation of the new curriculum! In teaching, as long as we insist on the wide application of vector method, let students master vector thinking method, make vertical and horizontal connection and extensive association with the help of vector, using contact viewpoint, motion viewpoint and aesthetic viewpoint, rationally reorganize and integrate mathematical knowledge and mathematical thinking method of each part, and fully demonstrate the process of applying vector; Reflecting the simple beauty and structural beauty of solving problems by vector method can fully reflect the teaching value of "vector" in improving students' mathematical ability. Through the statistics of the questionnaire, we can see that: 1, some students have no clear purpose for learning vectors, or have no clear learning purpose at all, which reflects that the front-line teachers in middle schools do not emphasize the educational value and significance, and do not emphasize the learning purpose, which leads to the blindness of students' learning. 2. Some students think that learning vectors is unnecessary and the original knowledge is enough, which is inseparable from the infiltration of teachers in the teaching process. They pay more attention to the application of traditional knowledge in solving problems, ignoring the powerful tool role of vector knowledge, which has not played its due role! 3. In the survey of students who have studied vectors, some students have a vague understanding of vectors, and think that it is good to simplify the learning burden in some aspects, but the pure dependence on vectors does not establish the correct concept of geometric three-dimensional, and the spatial imagination and three-dimensional quality cannot be fully developed. 4. Students' awareness of application is not strong. After learning new knowledge, it is not well integrated with previous knowledge, and knowledge becomes isolated, which is contrary to the comprehensiveness of mathematics and ignores the cultivation of creativity and analytical ability. Comprehensive analysis introduces vectors into high school mathematics textbooks, which requires students to master the basic theory and method of vectors. This is because vector knowledge has the following characteristics and requirements. First of all, using vectors to solve some mathematical problems will greatly simplify the original steps of solving problems with other mathematical tools, so that students can master an effective mathematical tool. Secondly, the introduction of vector will make a new analysis of the theory of "combination of numbers and shapes" in high school mathematics, and provide a brand-new method for implementing the teaching concept of "combination of numbers and shapes" in high school mathematics. Vector has a good "number-shape combination" characteristic. One is the form of "number", that is, a pair of real numbers can be used to express the size and direction of the vector at the same time; The second is the form of "shape", that is, a vector is represented by a directed line segment. Moreover, these two forms are closely related and can be transformed into each other through simple operations. It can be said that vectors are the best link between algebraic relations and geometric figures. It can quantify the relationship between graphs and algebra graphs, and liberate us from complicated graph analysis. We only need to study the vector relationship between these graphs, and we can get an accurate final conclusion. Make the analytical thinking and problem-solving steps concise and smooth, without losing rigor. Thirdly, the concept of vector itself comes from the study of physical quantities with both directions and sizes by the Department of Physics, which is called "vector" in physics. In fact, "vector" and "vector" are just two different names of the same quantity in mathematics and physics. In physics, vector is another important physical quantity relative to "scalar" with size but no direction. Almost all high school physics theories are explained by these two quantities. Vector is widely used in mechanics (such as force, velocity and acceleration). ) and electricity (such as current direction, electric field strength, etc. ), and introduce the vector chapter into the new textbook of senior high school to study and study vectors systematically and deeply. This will undoubtedly provide students with a mathematical foundation and many operational conveniences for learning and understanding vector knowledge in physics class. Similarly, students' physics practice related to vectors in physics class will make them have a deeper understanding of vectors and stimulate their interest and enthusiasm in learning vector knowledge. For example, in mechanics, the decomposition and synthesis of force and velocity use the theory of vector addition and subtraction, and the perfect combination of mathematics and physics also plays a similar role. Fourthly, introducing vector theory into high school textbooks is also a major trend of middle school education in the world today, which is the inevitable result of education adapting to the development of the times. The rise and development of trace vectors in mathematics is still a matter of recent decades. Looking through some early books about the history of mathematics, there are few introductions about the history of vector development. With the deepening of vector research, breakthroughs have been made in many aspects, and vector theory, like functions, trigonometry, complex numbers and other mathematical branches, has become more and more complete, forming an independent mathematical theoretical system. More and more mathematics educators realize that vector is not as abstruse as other emerging mathematics disciplines, and it is easy to be understood and accepted by students with high school education or above. Its good "combination of numbers and shapes" enables it to be integrated with high school mathematics knowledge and complement each other. Therefore, in order to keep up with the development of world mathematics education and make contemporary middle school students get in touch with the forefront of contemporary mathematics earlier, it is very necessary and feasible to introduce vectors into senior high school mathematics education. After introducing "vector" into high school mathematics textbooks, there are several problems worth discussing and thinking deeply. First of all, from the comparison between solving problems with vectors and solving problems without vectors, we can see that the advantage of solving problems with vectors is that a problem that can only be solved by complex analytic geometric analysis can be solved by simple deformation of vector formula. "This is the problem-solving mode of mathematics in the future, and it is also the progress of mathematics." Similarly, this idea is also the perfect embodiment of Descartes' mathematical philosophy of "turning practical problems into mathematical problems, then turning knowledge into equation problems, and then solving equations can solve problems". However, the front-line math teachers in senior high schools all know that it is one of the most important goals of senior high school math teaching to cultivate students' three abilities: calculation ability, analysis ability and spatial imagination ability. How can it be regarded as a kind of ability training for students to adopt such a simple solution that only needs to be substituted into formulas without any geometric analysis or even drawing? If students are only required to do such problems, they will be trained into "mathematical machines" that can only copy step by step and lack creativity, analysis and imagination. This is contrary to the training goal of contemporary mathematics. Secondly, most teachers who have been engaged in vector teaching will feel this way. That is, although the introduction of vector brings convenience to the derivation of other subsequent mathematical theories and the solution of difficult problems, its own theory and some problem-solving processes involved in it are difficult for students to understand and accept in the teaching process. This invisibly increases the teaching burden of middle school mathematics educators. Although it is very simple to solve some problems with this vector formula, it takes a lot of time for students to understand the origin and evolution of this formula. To solve this problem, the author thinks that in the final analysis, it is necessary to strengthen the careful teaching of vector knowledge and deepen students' understanding and flexible application of vector knowledge. Thirdly, for the introduction of vector chapters into new textbooks, the higher education authorities should also actively promote and train front-line teachers, and use policy instructions to intervene and guide when necessary to promote the smooth development of vector teaching in middle school teaching. But many middle school teachers have raised objections to adding vectors to high school textbooks, and they don't even understand them. There are two reasons: on the one hand, because the new textbook has just been implemented and everyone has no practical experience, it is difficult to find the advantages of vectors. On the other hand, many front-line teachers, especially old teachers, have been teaching old textbooks for many years, and their own vector knowledge and understanding of the advantages of vector teaching are relatively lacking. Therefore, in the process of popularizing new textbooks, it is very necessary to carry out short-term vector knowledge teaching training for mathematics teachers engaged in teaching new textbooks. In addition, the introduction and reasonable arrangement of a large amount of vector knowledge in the new textbook is also the most convincing evidence that educators and educatees think that vector knowledge should be taught well and learned well. With the deepening of teaching, the author's attitude towards vector has gone through a process from initial incomprehension to gradual understanding of its intention and essence, and finally recognition and earnest implementation in teaching practice. In addition, in middle school mathematics teaching, the phenomenon of ignoring vector chapters, passing by, and even not teaching or learning is also common in most schools. To fundamentally eliminate these phenomena, we need to rely on the correct guidance of educational reform.