Before class, the teacher prepares for this class by preparing lessons. Lesson preparation includes studying curriculum standards, studying teaching materials, reading reference materials, understanding and analyzing students' situation, determining specific and appropriate teaching methods, making lesson plans at different stages, and designing lesson plans for each class. The basic requirements for preparing lessons are as follows: 1. Learning textbooks, processing textbooks. Teachers and students are different in teaching materials, so we can't just stay in understanding, mastering and applying the conclusions. Because the compilation of teaching materials, in order to meet the requirements of simplicity and standardization, often compresses the formation process of concepts, covers up the discovery process of theorems, formulas and laws, hides the detailed elaboration process of mathematical ideas, and simplifies the refining process of laws. Teachers should not copy the basic knowledge from textbooks to lesson preparation books, but should deeply explore and analyze the formation process and rich connotation of basic concepts, and conduct extensive research and exploration on the discovery process of theorems, formulas and laws. It is necessary to investigate and conceive the cognitive process of mathematical laws in line with students' psychology. This is the work of teachers to process and recreate the teaching materials, and it is also the understanding and excavation of the teaching materials. Specifically solve the following problems: 1. Understand the basic requirements of teaching materials. The basic requirements of teaching materials include ideological content, the depth and breadth of basic knowledge, the level of basic skills and skills, and the focus of developing ability. The ideological content of teaching materials is mainly reflected in the dialectical materialism viewpoint contained in mathematics content. Many concepts in middle school mathematics have realistic models, so we should pay special attention to how these concepts are abstracted from concrete things in the real world when studying textbooks. The dialectical content in middle school mathematics content is particularly rich. For example, movement, change, development and transformation have changed from quantitative change to qualitative change, and the view of unity of opposites has almost penetrated into the specific content of each chapter. The mathematical historical materials involved in part of the teaching materials, especially the historical materials of Chinese mathematics, are also an important aspect of ideology. The depth and breadth of the basic knowledge contained in the textbook should not only be understood generally as a whole, but also analyzed carefully locally. For example, the basic knowledge of function, middle school mathematics is divided into four stages. There are two stages in junior high school: accumulating materials and establishing preliminary concepts. What are the specific requirements for each stage? How to master the discretion of each stage? Only on the basis of comprehensive investigation and thorough study of the contents, examples and exercises of each topic can these problems be truly clarified. The level of basic skills and skills requirements is mainly mastered through the study of examples and exercises. The emphasis of developing ability depends on the type and difficulty of mathematics content. For concept teaching, pay attention to the cultivation of observation, abstraction, generalization, discrimination and other abilities; For the teaching of theorems, laws and formulas, emphasis is placed on the cultivation of exploration ability such as induction, analogy, analysis and synthesis; For the teaching of easy mathematics content, pay attention to cultivating self-study ability; For the teaching of difficult content, we should pay attention to cultivating the ability to analyze and solve problems, and so on. 2. Exchange knowledge and master the knowledge system of teaching materials. Middle school mathematics is a systematic and rigorous subject. Mathematics teachers should not only delve into the basic knowledge of the textbook, but also be familiar with the position and role of the basic knowledge taught in the whole mathematics textbook, the relationship between the basic knowledge and the gradual change of the knowledge structure with the deepening of learning, which requires teachers to have the ability to grasp the knowledge system. For example, when studying teaching content, teachers should pay great attention to the problem of "how to introduce the old and bring the new". Only by attaching importance to the relationship between knowledge can students learn knowledge in isolation, not just by memorizing it. There are many relationships between knowledge, including subordination, juxtaposition, development and deepening, correspondence and so on. Teachers' attention to the relationship between knowledge can be explored from these relationships. 3. Determine the key points, difficulties and keys Generally speaking, the content that runs through the whole situation, is comprehensive, is widely used in learning, plays a central role in students' cognitive structure, and plays a basic role and a link role in further learning is the focus of teaching. This is determined by its position and role in the knowledge structure of textbooks. Usually, definitions, theorems, formulas and rules in textbooks are the focus of teaching, the focus of teaching mathematical thinking methods, the training requirements of basic skills, and the thinking process of deducing theorems and formulas are the focus of teaching. Teaching difficulty is the differentiation point of students' academic performance. Generally speaking, knowledge is too abstract, the internal structure of knowledge is too complex, the essential attributes of concepts are relatively hidden, and new perspectives and methods are needed to study knowledge from old to new, and all kinds of inverse operations are the factors that cause difficulties. It is a very complicated job to analyze the difficulties in teaching. Teachers should consider and comprehensively analyze the characteristics of teaching materials, the contradictions in the teaching process and the psychological obstacles of students' learning. The key in teaching refers to the knowledge content that can play a decisive role in mastering a certain part of knowledge or solving a certain problem. If you master this part of knowledge, the rest will be easy to master, or a little patience will solve the whole problem. 4. The material thinking of providing students with thinking training needs to be realized by analyzing, synthesizing, comparing, abstracting and summarizing the teaching contents. Providing thinking training materials is the processing and organization of teaching content to train students' thinking. Thinking training materials are divided into two categories: one is perceptual materials, which train thinking with perceptual representations; The other is rational material, which trains thinking with basic thinking forms such as concept, judgment and reasoning. When students solve new problems, their thinking is often blocked. Teachers need to find connection points to solve new problems in students' original cognitive structure and provide materials to reproduce old knowledge. It is difficult for students to accept new knowledge and skills, and the teaching content has a certain leap. Teachers need to arrange paving materials, introduce the new with the old and the high with the low, and reveal the logical relationship between the old and the new. In order to cultivate students' ability, teachers need to provide exploratory thinking training materials and design various variant training questions. 5. Correctly preparing and organizing exercises is an integral part of mathematics teaching, which is very important for students to master basic knowledge, basic skills and develop their abilities, and is a necessary condition for learning mathematics well. The purpose of the exercise is to make students further understand and master the basic knowledge of mathematics, train, cultivate and develop students' basic skills and abilities, find and make up for the omissions and deficiencies in learning in time, and cultivate students' good study habits and quality. We should pay attention to give full play to the role of exercises, strengthen the correct guidance of solving problems, and guide students to make necessary generalizations from the thinking methods of solving problems. Teachers must strengthen the research on exercises and carefully select and arrange exercises. The choice must be considered from the aspects of purpose, content, form, weight and students' acceptance ability, so that students can practice correctly and effectively. In order to make the topic meet the above requirements, teachers should first calculate all the exercises in the textbook twice according to the requirements, so that students can solve the following problems. (1) Define the purpose and requirements of the exercise. For three different types of exercises, exercises and review reference questions in the textbook, we should pay attention to the specific requirements, key problems, problem-solving skills and exercise format of each question, and analyze which students can complete independently, which need tips and which need explanations and demonstrations. (2) It is clear that the key exercises of the exercises serve to consolidate the basic knowledge. Therefore, when choosing exercises, we must also consider the characteristics of knowledge and students' acceptance ability, so that students can concentrate on training that is conducive to developing intelligence and mastering basic knowledge and skills, and pay attention to which are primary and which are secondary, so as to grasp the key points of exercises in class exercises and homework. (3) Determining the problem-solving methods of exercises requires students to solve problems in various ways, which can improve students' interest in learning and cultivate students' problem-solving ability in many ways. According to the different characteristics of teaching requirements and topics, as well as the specific situation of students' acceptance ability and intellectual development level, teachers should practice through oral answers, blackboard writing performances, examination of topics, written homework and thinking. Generally speaking, the number is simple, the operation is not complicated or the argument is easy, which is a question that needs to be answered by mastering a basic concept, theorem, law or formula, so it can be used as an answer; Calculation or demonstration is not complicated, but it is typical, which can reflect the concrete application of knowledge and skills and the standardization of writing format, and can be used as blackboard writing performance; Exercises that can consolidate old knowledge and easily introduce new curriculum content from it should be reviewed and asked in class; Exercises with complex calculation or difficult demonstration involving a wide range of knowledge can be used as written homework questions; The exercises with strong thinking and complicated narration should be thinking questions. (4) To measure the weight of exercises, we should determine the weight of exercises according to the difficulty of the topic and the ability of students to solve problems. The topic is too simple, the weight is too small, and it is easy to complete the task of solving problems, which not only fails to achieve the purpose of practice, but also makes students feel complacent. The problem is too complicated and heavy, and students can't finish the task within the specified time, which will not only make students lose confidence, but also increase the burden and affect their all-round development. Generally speaking, the amount of exercises assigned by teachers to students should be determined according to the possible practice time in and out of class and the speed ratio of teachers and students to solve problems (1: 3- 1: 4). However, due to the uneven level of students, attention should be paid to teaching students in accordance with their aptitude when assigning homework. In addition to unified basic exercises, there are also some demanding selected exercises or thinking questions to meet the needs of students who study well and make their mathematics develop. In addition, teachers should be good at compiling, adapting or selecting some supplementary questions according to the needs of teaching materials and students, especially some transitional questions, contact questions and comprehensive questions, so that students can better understand the content, master the methods and use them flexibly. Second, the understanding and analysis of students' situation. Teachers always ask students to achieve certain goals within a certain period of time. Whether this goal can be achieved requires teachers to understand the students' learning status quo and find out the gap between the status quo and the goal and the reasons for the gap. 1. Understanding and analyzing students' cognitive structure refers to the organization of the essence of subject knowledge in students' minds. In the process of learning, whether students' original cognitive structure has appropriate ideas for absorbing new knowledge, whether these ideas are stable and clear, and the degree of distinction between new knowledge and knowledge in students' existing cognitive structure will directly affect the effect of learning activities. For example, in the seventh grade "Operation of Rational Numbers" teaching, new knowledge includes: addition and subtraction of rational numbers (operation rules and algorithms), multiplication and division of rational numbers (operation rules and algorithms). In the original cognitive structure of students, the concept of assimilating new knowledge is not only the knowledge of rational numbers, but also the arithmetic numbers and four operations of arithmetic numbers in primary schools. Therefore, the focus of rational number teaching is the analysis of students' cognitive structure, which will pay attention to what students have understood and mastered in classroom teaching, and teachers do not need to explain more; What students know, but with the deepening of mathematics learning and the improvement of content requirements, teachers need to guide them; What content is not well understood and mastered, but the teacher can deepen the understanding and mastery with a little help; What knowledge is still difficult, which needs to be carefully arranged by teachers in the future teaching and solved step by step. 2. Understand the students' thinking state and thinking characteristics. Experienced teachers should not only analyze the results of students' learning, but also take the process of producing the results as the focus of learning analysis, attach importance to the psychological activity process and psychological change law of students' learning mathematics, and attach importance to analyzing various obstacles of students' learning psychology. In order to eliminate obstacles in time, teachers should analyze the causes of psychological obstacles. Many students may have done the same topic correctly, but they may have different thinking processes; Some students may have made mistakes, but the reasons may be different. It is also necessary to analyze the personality and psychological characteristics of different types of students, because each student's physical quality, living environment, educational conditions and hobbies are different. It is necessary to study the learning psychology of spare students and analyze the personality psychology of students with learning difficulties. The understanding and analysis of students' learning provides a basis for teachers' teaching design. After mastering the above situation, students in the whole class can generally be divided into three categories according to excellent learning, average learning and learning difficulties. The design of classroom teaching is based on the situation of most middle-level students, while paying due attention to both ends. Teachers can learn about students' learning from the following aspects. First, establish a teaching feedback system to let teachers know the learning situation in time and effectively control teaching. This requires that in classroom teaching, teachers should not only output information, but also pay attention to some subtle reactions of students in classroom learning. Correcting homework after class is not a routine, but an important source of receiving teaching information. Second, dredge various channels to understand the learning situation. Teachers must contact students extensively, listen to their answers, don't deny students' narratives easily, and give students more opportunities to express their ideas. Third, take the initiative to open up ways to understand the learning situation. Establish students' learning files and systematically understand students' knowledge base; Understand students' personality, psychology and hobbies through other teachers; Home visits, individual counseling, heart-to-heart, targeted understanding of students, and so on. Third, the significance of determining the purpose of classroom teaching mainly includes four aspects: ① Teaching purpose is the basis for choosing teaching content and teaching methods; (2) The purpose of middle school mathematics teaching stipulated in the curriculum standard is realized through the teaching purpose of each chapter and classroom; ③ The teaching purpose is the basis for evaluating the teaching effect; (4) The teaching purpose provides a basis for students to make clear their learning requirements. If the teaching purpose of each class is vague, it is difficult to arouse students' attention and interest in teaching, let alone let students learn independently in class. When making teaching objectives, we should pay attention to the educational objectives of cultivating knowledge, skills and abilities, ideological and moral concepts, non-intellectual factors and personality development. These three angles are planned organically and cannot be neglected. The teaching of a certain class may emphasize several aspects, but from the whole chapter and the whole course, the overall purpose should be the best. In addition, based on students' learning foundation and possible development level, we should formulate teaching objectives and summarize them in concise language. Iv. Selection and organization of teaching contents and methods The selection and organization of teaching contents is to process the teaching materials into procedural materials that facilitate bilateral activities between teachers and students in the classroom according to the requirements of teaching purposes. The materials here are not limited to a textbook, but also include teaching reference books and instructions. Teachers should, according to the actual situation and needs, focus on the teaching purpose, supplement appropriate materials, quote classics, solicit quotations extensively, enrich classroom content and improve teaching efficiency. When choosing and organizing teaching content, teaching methods are actually considered at the same time. Generally speaking, any class will not adopt a single teaching method, but often adopt a combination of various teaching methods. In the process of teaching, there is not a one-to-one correspondence between teaching methods and teaching contents. For the same teaching content and purpose, the choice of teaching methods can vary from person to person and from place to place. Even if the content is the same, the students are the same and the teaching conditions are the same, we can choose different methods to achieve good teaching results. Various teaching methods have their own characteristics and functions. According to the basic forms of students' cognitive activities under the guidance of teachers, teaching methods can be divided into four types. 1. If the teaching method is not used properly, it will lead to "full house irrigation" in teaching. When using teaching methods, we should pay attention to the following aspects: (1) The teaching content should be highly scientific, ideological and systematic, and we should grasp the logical structure of the content and grasp the key points and keys. (2) In teaching, students should be inspired to think positively based on the original cognitive structure, and teachers should transform the process of asking questions, analyzing problems and solving problems into students' cognitive process. (3) In the teaching process, we should be good at creating problem situations, that is, be good at setting doubts and solving doubts. (4) The teaching language should be accurate, concise, clear, vivid, easy to understand and at a moderate speed. 2. Discussion Research The general mode of discussion research method is: (1) Teachers create problem situations and ask questions; (2) Students think independently about their own problems and sort out their own analysis ideas and solutions; (3) Students discuss research in small groups or large groups, exchange their own thinking achievements, that is, the advantages and disadvantages of non-simplification and conception, and debate and discuss the proposed solutions from different angles; (4) Teachers debate and discuss. Quantitative encouragement of students' positive thinking and qualitative evaluation can change students' passive learning state, promote the multi-directional information exchange between teachers and students and form a lively learning atmosphere. The disadvantage of discussion research method is that there is little discussion content, which is not conducive to the all-round cultivation and improvement of ability. If students' learning foundation is uneven, discussing problems or becoming a mere formality or getting into trouble, the purpose of discussion and research will not be achieved. In addition, more time is needed to discuss the research methods. 3. Self-study method Self-study method focuses on "learning". The general model is: (1) teachers ask questions, arrange contents, list outlines and guide self-study. (2) Students read the content and practice to find problems. (3) Teachers patrol to understand and master the progress and difficulties of students' self-study, starting from their own foundation. Individual guidance (4) Teachers give detailed lectures to deeply understand the content, highlight key points, solve difficulties and summarize laws (5) Students practice self-study at a higher level by using knowledge. The disadvantage is that students with poor foundation are more difficult to adapt, lack of mutual encouragement of intelligence, and the learning atmosphere is rather dull. 4. The general model of discovery method is as follows: (1) Teachers create problem situations, put forward problems that need to be solved or studied, trigger students' cognitive conflicts, stimulate the requirements of inquiry, and clarify the goals or centers of discovery; (2) Put forward the hypothesis to solve the problem, guide students to think and choose various solutions to solve the problem; (3) Help students prove hypotheses. If there are different opinions, students can use their own knowledge to explain their views and put forward arguments and arguments. (4) Teachers summarize arguments and proofs, draw the same conclusion, and feedback and consolidate them in time, so that students can establish a new cognitive structure. How to choose or determine a good teaching method must consider the following three main factors: teaching purpose, teaching content and teaching object. In addition to these three main factors, teachers' own conditions and teaching materials are also factors that need attention.