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; The simple beauty and structural beauty of 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 don't have a clear purpose for learning vectors, or they don't have a clear learning purpose at all, which reflects that the front-line teachers in middle schools don't emphasize the educational value and significance, and they don't 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 relying solely on vectors has not established the correct concept of geometric three-dimensional, and the spatial imagination and three-dimensional quality can not 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.
Through comprehensive analysis, vector is introduced into high school mathematics textbooks as a basic theory and method, which requires students to master. 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 the good characteristic of "combination of numbers and shapes". 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 directed line segment is used to represent a vector. 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 graphics, algebra the relationship between them, and liberate us from the complicated graphic analysis. We only need to study the vector relationship between these figures, and then we can draw an accurate final conclusion and make an analysis.
Thirdly, the concept of vector itself comes from the study of physical quantities with both directions and sizes by the physics department, 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 high school textbook. Systematic and in-depth study and research on vectors will undoubtedly provide students with mathematical foundation and many operational conveniences for learning and understanding vector knowledge in physics class. Similarly, the physical reality related to vectors that students encounter in physics class will give them 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 traceability vector in mathematics is still something in 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 vectors are not as abstruse as other emerging mathematics disciplines. It is easy for students with high school education or above to understand and accept, and its good "combination of numbers and shapes" makes it integrate 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, several problems worth discussing and thinking deeply.
First of all, from the comparison between the method of solving problems with vectors and the method of 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 simply by using the simple deformation of vector formula. "This is the problem-solving model of mathematics in the future and the progress of mathematics." Similarly, this idea is also an embodiment of Descartes' "turning practical problems into mathematical problems, and then learning variables is called equation problems". Then the problem can only be solved by solving the equation, which is the perfect embodiment of mathematical philosophy. 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 and does not need any geometric analysis or even drawing in the process of solving problems? If students are only required to do such problems, they will be trained into "mathematical machines" that only plagiarize step by step and lack creativity, analysis and imagination. This runs counter to the training goal of contemporary mathematics.
Secondly, most teachers who have been engaged in vector teaching will feel that although the introduction of vector has brought 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, which invisibly increases the teaching burden of middle school mathematics educators. Although the method of solving some problems with this vector formula is simple, 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 introducing vector chapters into new textbooks, 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, 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. It can be seen that in the process of promoting the new textbook, it is quite necessary to conduct short-term vector knowledge teaching training for mathematics teachers engaged in the teaching of the new textbook. 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 feel 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.