Teaching principles are the reflection of teaching laws and the crystallization of teaching experience, the basic requirements for guiding teaching work and the basic principles that teachers must abide by in teaching work.

The general teaching principles defined by China's educational circles are: the principle of combining science with ideology, the principle of combining theory with practice, the principle of combining teachers' leading role with students' consciousness and enthusiasm, the principle of combining perception with understanding, the principle of gradual and systematic progress, the principle of consolidating knowledge and skills, the principle of conforming to students' age characteristics and acceptance, and the principle of unifying requirements and teaching students in accordance with their aptitude.

Under the guidance of general teaching principles, due to the particularity of teaching in various disciplines, the teaching of various disciplines should also follow the discipline teaching principles that conform to the characteristics of disciplines and the age characteristics of students.

In the era of knowledge transfer, the majority of mathematics educators and teachers in China have summed up many effective principles of middle school mathematics teaching according to the characteristics of middle school mathematics, teaching experience and the age characteristics of middle school students, among which the most influential ones are: the principle of combining rigor with ability, the principle of combining abstraction with concreteness, the principle of combining theory with practice, and the principle of combining consolidation with development.

First, the principle of combining rigor with competence

1. The rigidity of mathematical theory

Stiffness is one of the basic characteristics of mathematical science theory, and its meaning mainly refers to the rigor of mathematical logic and the accuracy of conclusions, and it is no exception in middle school mathematical theory. Mainly in the following two aspects: first, we must define the concept (except the original concept); Second, propositions (except axioms) must be proved. Therefore,

(1) The mathematical concepts contained in each branch of mathematics are divided into two categories: original concepts and definition concepts. The original concept is the starting point of defining other concepts in this discipline, and its essential attributes cannot be expressed by definition in this discipline, but can only be revealed by axioms. The concept of definition must be precise and logical.

(2) The true propositions contained in each branch of mathematics are also divided into axioms and theorems. Axiom is selected as the original basis to prove the correctness of other true propositions in this discipline, and its own correctness is recognized without logical proof. However, as a system, they must meet the requirements of compatibility (no contradiction), independence and integrity. Theorem must be proved by logic.

(3) The concepts and true propositions of each branch of mathematics form a system in a certain logical order. In this system, each defined concept must be defined by a previously known concept; Every theorem must be derived from a proposition whose correctness is known.

(4) The presentation of concepts and propositions and the process of argumentation of propositions are becoming more and more symbolic and formal.

But the rigor of mathematics is relative and develops gradually. Fossilization does not exist in the early stage of the development of various branches of mathematics, and can only be achieved in the final stage of perfection. For example, the concept of function has gone through seven stages of development before it is gradually rigorous. Euclidean geometry didn't really become rigorous until the establishment of Hilbert axiom system at the end of 19. There is another aspect of the rigor of mathematics. For example, basic mathematics focuses on theory and applied mathematics focuses on application.

2. The ability of middle school students

In mastering the rigor of mathematical science, we must do what we can according to the knowledge level and acceptance ability of middle school students. We should pay attention to the following points:

(1) The strict requirements in mathematics can only be gradually adapted, and middle school students can gradually achieve them in the learning process from lower grades to higher grades. When they first start learning, they are often not rigorous enough. Understanding depends on intuition and solving problems depends on imitation. For example, the idea of set and correspondence permeates the mathematics textbooks of primary and junior high schools, but it is not until the high school stage that the preliminary research is carried out and the rational understanding stage is entered that the strict requirements can be gradually reached. Therefore, in teaching, we must conform to the development law of students' cognition, put forward requirements appropriately, do what we can, improve requirements step by step in a planned way, and gradually understand and master the requirements of teaching rigor.

(2) The understanding of mathematical rigor is relative. Because the rigor of mathematics is relative, it takes a long time for human beings to understand the rigor of mathematics. Moreover, middle school students' learning is also a cognitive activity. Learning mathematics is to realize the achievements made by human beings after a long history. This cognitive process is not necessary and impossible to repeat history, but under the guidance of teachers. It follows the general law of understanding from low level to high level, from simple to complex, from shallow to deep, and gradually deepens. In addition, middle school math class hours and students' original basic knowledge and ability are limited, so middle school students can only know the most basic contents and methods of mathematics. Correspondingly, their understanding of mathematical rigor can only be basic, relative and preliminary.

(3) The intellectual development of middle school students has strong plasticity. Middle school is a period of rapid development of teenagers' intelligence. Middle school students' ability to accept knowledge is limited and plastic, so their cognitive potential should be fully estimated. In teaching, we should properly mobilize students' enthusiasm, give full play to their potential and promote their thinking development.

3. Combination of rigor and ability

Mathematical science is rigorous, and middle school students' understanding of mathematical science is restricted by the principle of ability. Therefore, in mathematics teaching, it is the general requirement of the principle of combining rigor with ability to reflect the true nature of mathematics science and conform to the reality of students. The essence of this principle is that both rigor and ability should be taken into account in mathematics teaching. On the one hand, appropriate and clear goals and tasks should be put forward for each stage of mathematics teaching, on the other hand, students' logic should be cultivated step by step.

In mathematics teaching, the principle of combining rigor with ability is mainly realized through the following requirements.

(1) Teaching requirements should be appropriate and clear. That is, according to the principle of combining rigor with ability, the relationship between scientific mathematics system and mathematics system as a middle school education discipline should be properly handled.

(2) Teaching should be rigorous in logic, clear in thinking and accurate in language. In other words, when explaining mathematical knowledge, we should consciously infiltrate the knowledge of formal logic, pay attention to cultivating logical thinking, and learn to reason and demonstrate. Every noun, term, formula and law in mathematics has an accurate meaning. Whether students can accurately understand their own meaning is one of the important signs to ensure the scientific nature of mathematics teaching, and the degree of students' understanding is often reflected in their language.

In order to cultivate students' language accuracy, teachers should have high mathematical language literacy. New teachers should overcome two tendencies in language: First, they abuse languages and symbols that students can't accept. For example, they want to tell junior one students that "the judgment nature contained in the definition of each concept is necessary and sufficient" and use the double arrow symbol to indicate it. The second is to bring everyday popular but inaccurate idioms into teaching. For example, when teaching score reduction, they often say, "Go to the top."

Therefore, the language of mathematics teachers should be concise and accurate, and strive to meet the requirements of standardization. In order to prevent the phenomenon of arbitrary definition and arbitrary judgment in teaching, we should not replace mathematical terms with vague life terms just for the sake of being easy to understand.

(3) In teaching, pay attention to explaining mathematics knowledge from the shallow to the deep, from the easy to the difficult, from the known to the unknown, from the concrete to the abstract, from the special to the general, and be good at stimulating students' curiosity, but the problems involved should not be too difficult to make students daunting, so as to achieve good teaching results.

In short, while emphasizing rigor, students' acceptability cannot be ignored; When emphasizing ability, we should not ignore the content of science. Only by organically combining the two can the teaching quality be improved.

2. The principle of combining abstraction with concreteness

1. Mathematical abstraction

All science is abstract, but mathematics is the abstraction of the spatial form and quantitative relationship of objective things, and it is one of the most common and essential characteristics of things. Therefore, the abstraction of mathematics needs to abandon all other characteristics of things and achieve a high degree of abstraction.

The abstraction of mathematics also has the characteristics of high universality and wide application. Generalization is the thinking process of extending an attribute abstracted from some objects to similar objects. For example, a problem-solving method abstracted from solving a certain kind of exercises is extended to solving similar exercises. Abstraction and generalization are interrelated and inseparable. The higher the abstraction of mathematics, the stronger its universality and the wider its application scope.

The abstraction of mathematics is also manifested in the extensive and systematic use of mathematical symbols, which has the characteristics of trinity of words, meanings and symbols, which is incomparable to other disciplines. For example, the word "parallel" has special positional relationships among straight lines, straight lines and planes, and planes and planes, which are represented by special symbols "//"and can be represented by concrete figures.

Mathematical abstraction is a process of gradual abstraction, gradual improvement and re-abstraction. If we pay full attention to this feature in mathematics teaching, we can effectively cultivate students' abstract generalization ability.

2. The limitations of students' abstract thinking

Middle school students are at the level of image thinking and experience abstract thinking, and gradually transition to theoretical abstract thinking in high school. Due to the influence of age, the ability to understand problems and the orientation of understanding problems, their abstract thinking has certain limitations, which are as follows: excessive dependence on specific materials, that is, abstract examples can be given for the concepts and conclusions they have learned; Concrete and abstract are separated, and the understanding and mastery of abstract theory is one-sided and limited, so abstract theory can not be applied to concrete problems; The relationship between abstract mathematical objects is difficult to grasp.

3. Combination of abstraction and concreteness

The abstraction of mathematical theory and the limitation of middle school students' abstract thinking are a pair of contradictions in middle school mathematics teaching. How to deal with this contradiction lies in correctly understanding the basic relationship between concreteness and abstraction-concreteness is the foundation of abstraction, abstraction is the destination of concreteness, and concreteness needs to be promoted to a higher level of abstraction.

(1) From concrete to abstract, cultivate and develop students' abstract thinking ability and innovative consciousness. From concreteness to abstraction is a leap in understanding, and it is a stage in which sensibility rises to rationality. In middle school mathematics teaching, we should pay attention to introducing examples to form intuitive images and providing perceptual materials through intuitive objects (including teaching AIDS), which is an effective way to promote and develop students' abstract thinking ability. By observing the relationship between the intersection line between walls and the intersection line between walls and the ground in the classroom, the concept of out-of-plane vertical line is introduced. Attention should be paid to introducing from special cases and explaining general laws. For example, when solving a quadratic equation with one variable, we usually learn x2=a type first, then (x+a)2=b type, and then learn ax2+bx+c=0 type, which is easier for students to accept.

In middle school mathematics teaching, in order to cultivate and develop students' abstract thinking ability, teachers' main task is to create specific mathematics situations, inspire and guide students to actively participate in teaching activities and prevent arranged substitution.

(2) Form skills from abstract to concrete, and further cultivate students' ability to analyze and solve problems. From abstraction to concreteness is another stage of cognition and another leap from concrete perceptual knowledge to abstract rational knowledge. It belongs to a more important stage in the whole cognitive process, that is, a new stage of applying mathematical theory to solve problems initially and concretizing rational knowledge.

From abstract to concrete, it is to let students solve concrete practical problems on the basis of mastering abstract mathematical theory, so as to prepare for further from concrete to abstract. The process of solving mathematical problems is mainly the application of abstract mathematical theory, the process of forming mathematical related skills, and the process of further cultivating and developing observation ability and logical thinking ability such as analysis and synthesis. When solving mathematical problems, we may learn some new mathematical ideas and methods besides using abstract theory, which also plays a certain role in cultivating students' creative thinking ability.

The combination of abstraction and concreteness is to make students understand abstract theory correctly and profoundly. Concrete and intuition are just means, and the fundamental purpose is to cultivate abstract thinking ability. Therefore, only by constantly implementing the process of concreteness-abstraction-concreteness and repetition can we continuously develop our learning in depth and gradually improve and deepen our understanding.

Three. The principle of combining theory with practice

1. Dialectical Unity of Mathematical Theory and Practice

The abstractness and preciseness of mathematical theory have a practical basis, and mathematical theory has a wide range of applications. This shows that mathematical theory not only comes from practice, but also guides practice and is tested and developed in practice. This is the dialectical unity of mathematical theory and practice.

Mathematical theory comes from practice. Through the analysis and synthesis of various objective things and phenomena in practice, simple and universal truths are summarized, thus forming abstract theories, which is the cognitive process of "from complex to simple". For example, the quadratic function y=ax2 is an abstract generalization of many actual quantitative relations. After forming the abstract theory of this mathematical model, it is more universal. By giving different meanings to the letters, we can express different quantitative relations, such as the formula of free fall S=gt2, the formula of energy E=mv2, the formula of circular area S=πr2, and so on.

It is precisely because of the simplicity and universality of mathematical theory that it can be used to "control complexity with simplicity", guide practice and solve problems extensively, and at the same time test and develop the theory in the practice of solving problems.

2. The current situation of middle school students learning mathematics

The process of middle school students' learning mathematics is a special process of understanding and practice, which is a learning process with classroom teaching as the main form and indirect knowledge as the main form under the guidance of teachers.

The knowledge of mathematical theory learned by middle school students is formed by the practice of predecessors for centuries. Due to the limited classroom teaching time, it is impossible and unnecessary to proceed from reality and let students discover everything. However, students should try their best to understand the actual background and context of knowledge, participate in the formation process of knowledge, and gradually establish a correct view of mathematics.

It is difficult for middle school students to abstract practical problems in production and life into clear mathematical problems, so as to establish clear mathematical models, which is also an important reason why many students are afraid and unwilling to learn mathematics.

Because middle school students don't know or understand the mathematical principles deeply and are not good at concrete analysis, they often stay at the level of rote memorization and mechanical copying, and often don't analyze the quantitative relationship in mathematical problems clearly. Therefore, it is difficult to give play to the guiding role of theory in solving practical problems.

3. Combining theory with practice

The combination of theory and practice is not only the basic principle of epistemology and methodology, but also the basic principle of teaching theory and learning theory. When applying this principle to teaching, we should pay attention to the following aspects:

(1) Pay attention to the connection between middle school mathematics and practice. In teaching, teachers must proceed from reality, from the familiar life and production practice of students, create appropriate mathematical situations, and gradually teach students to put forward and solve mathematical problems, so as to gradually achieve the unity of mathematical knowledge and practice.

(2) Vigorously improve the theoretical level and strengthen the guiding role of theory. The central link of integrating theory with practice is to deeply understand the theory and give full play to its guiding role. Only by deepening the understanding of knowledge and improving the theoretical level of mathematics teaching in middle schools can we firmly grasp the relevant mathematics knowledge and apply it to practice. An important reason for the great influence of exam-oriented education is that the theoretical level is not high, there is no theoretical guidance, and only algorithms are emphasized, but arithmetic is not emphasized. Do not pay attention to understanding and systematic mastery, but be satisfied with memory and imitation; Do not pay attention to scientific "general methods", pursue so-called problem-solving skills and so on.

(3) Grasp the degree of combining theory with practice. How to create mathematical situation in middle school mathematics teaching, so that it is closely related to the mathematical knowledge to be learned, which is helpful to cultivate students' questioning ability; What typical practical problems students should master, what degree and requirements mathematical problems should reach according to the mathematical situation, and how to train students' abilities of abstraction, analysis, synthesis and analogy by abstracting mathematical problems from practical problems according to the thinking method of mathematical modeling, so as to establish mathematical models and solve mathematical problems in a planned, regular and comprehensive way need to be considered.

Four. The principle of combining consolidation with development

The combination of consolidation and development is one of the scientific teaching principles, which is determined by the curriculum objectives, teaching characteristics and laws of middle school mathematics and restricted by the psychological laws of human memory development. Consolidation is the development of knowledge, which in turn can promote the firm mastery of knowledge.

1. Consolidate mathematics knowledge.

The mastery of knowledge includes four related levels and processes: perception, understanding, consolidation and application. Perception is from ignorance to knowledge, comprehension is from superficial knowledge to profound knowledge, consolidation is from forgetting to keeping, and application is the process of realizing action. The purpose of mastering knowledge is to apply it, but if the learned knowledge is not consolidated enough, the application will become empty talk. The key to consolidating what you have learned lies in memory. Only by improving memory can we firmly master the basic knowledge and skills of mathematics.

(1) Understanding is the basis of memory. Mathematical knowledge can only be firmly remembered on the basis of profound understanding. In teaching, strengthening the teaching of basic knowledge, revealing the essence of mathematical facts, mathematical concepts and principles from many aspects, and establishing a certain logical system to make students deeply understand are effective ways to enhance their memory and consolidate their knowledge. Being good at guiding students to understand the relationship between things, making full use of existing knowledge and experience, making new connections on the basis of existing connections, bringing new knowledge into the corresponding knowledge system, and constantly enriching and perfecting the cognitive structure are also good ways for students to deeply understand and firmly remember.

(2) The organic combination of image memory and logical memory. In teaching, fully revealing the relationship between mathematical knowledge and objective reality, the relationship between old and new knowledge and the internal relationship between units, and combining theoretical knowledge with practice by means of visualization are conducive to consolidating knowledge. Therefore, in addition to paying attention to logical reasoning, we should also pay attention to the use of appropriate visual means, such as physical objects and laws.

(3) Through induction and analogy, association can promote memory. For similar things with similar properties and shapes, similar associations can be caused. Contrastive association caused by things with opposite characteristics can cause association on the other side of contradiction when one side of contradiction appears, thus improving the memory effect. Relational association can also be carried out from the causal relationship and subordinate relationship of things. For example, the concept of number is expanded, and its knowledge content is logically related.

(4) The combination of memory and reproduction accelerates and consolidates memory. In teaching, students should master the law of forgetting, organize review reasonably and strive to promote the reproduction of knowledge. At the same time, we should pay attention to the diversification of review methods to prevent monotonous mechanical repetition, so as to improve the efficiency of consolidating knowledge.

2. Pay attention to developing students' thinking.

The purpose of mathematics teaching is not only to make students master systematic knowledge and skills, but also to cultivate students' innovative thinking and practical ability. Only by developing students' thinking can we understand and consolidate what we have learned more deeply, thus improving students' practical ability. "Mathematics is the gymnastics of human thinking" shows that mathematics teaching must develop students' thinking and be beneficial to students' thinking.

(1) Clarify the goal and thinking direction in teaching. Students' thinking begins with questions, and there are no challenging questions that can't stimulate students' thinking. Therefore, in teaching, we should ask enlightening questions, create problem situations, make students clear their thinking direction, so as to stimulate their interest in learning, promote the development of thinking, put forward mathematical problems, and then solve them so that students can apply them to practice.

When talking about the classification of triangles, a teacher gave the following three pictures.

Ask the students to judge the type of triangle according to the part of the triangle clearly shown in the picture. When students judge the type of triangle in the first picture, there is a great debate. Finally, under the guidance of the teacher, the understanding was unified, and the correct results were obtained, which promoted the development of students' thinking.

(2) Provide students with sufficient raw materials for thinking and processing. Students' thinking process is the process of processing input information, so information is the raw material of thinking processing. With sufficient raw materials, thinking processing can be carried out effectively. In middle school mathematics teaching, the information that can be provided to students is nothing more than language and representation. Mathematical formulas and symbols belong to linguistic information, while images, models and teaching AIDS belong to expressive information. In teaching, only constant enrichment and accumulation can be achieved.

(3) Develop abstract thinking forms. To develop the thinking form, we must develop the thinking form. There are three forms of abstract thinking: concept, judgment and reasoning. Concept is the foundation, judgment is the connection of concepts, and reasoning is the combination of judgments. In middle school mathematics teaching, students should first master a series of mathematical concepts in order to make correct judgments and inferences on this basis. Only in this way can they continue to master basic mathematical knowledge and certain mathematical skills.

(4) Teach students to master thinking methods. The thinking methods in middle school mathematics generally include: analysis and synthesis, comparison and classification, abstraction and generalization, induction and deduction, systematization and concretization, generalization and specialization, etc. These thinking methods are interrelated and intertwined. In the practice of learning and application, we must use it comprehensively to think normally, understand and consolidate what we have learned, and find and solve problems in practice.

3. Combination of consolidation and development

The combination of consolidation and development is to firmly grasp the basic knowledge and skills of mathematics, develop thinking and improve ability. The key to consolidating knowledge lies in the systematization and application of knowledge, and the key to developing thinking lies in logicalization and training. Therefore, in teaching, we should effectively organize review, review the old and learn the new, and draw inferences from others, so that students' knowledge can be systematized and deepened, their thinking can be trained and developed, and their ability can be improved.

In order to implement the principle of combining consolidation with development in teaching, we should pay attention to the following two aspects:

(1) Seriously study and review the work of consolidating the knowledge, skills and methods that students have learned. It is necessary to review the basic knowledge comprehensively and systematically so that students can understand the basic mathematical ideas and methods. It is necessary to review the unit and the general review in time to systematize the knowledge learned and form an organic knowledge system. If we understand the mathematical ideas and methods in the knowledge system, we can not only draw inferences from others and use them flexibly, but also achieve the goal of consolidation and deepening.

(2) Focusing on the teaching purpose, focusing on developing thinking and cultivating ability, carefully selecting review questions. The selection of review questions should not only be conceptual, basic, typical, targeted and comprehensive, but also enlightening, reflective, flexible and creative. For example, reviewing with complete sets of questions is conducive to mobilizing various means and connecting with various methods to improve students' ability to apply mathematical knowledge; Reviewing the exercises with multiple solutions to one problem is conducive to developing students' thinking of seeking differences and improving their ability to solve problems; Using variant questions to review is conducive to cultivating students' flexibility and creativity in thinking; Revising mistakes is conducive to cultivating students' critical thinking and improving their scientific discrimination ability; Reviewing with extended questions can cultivate the flexibility and profundity of students' thinking and improve their mathematical ability.