First, guide students to start with his natural ideas about problems, raise life experience to mathematical concepts, and gradually connect with formal mathematical knowledge.
Mathematics comes from life, it is concrete, but it is abstract. Formalization is an inherent feature of mathematics and an important part of rational thinking. Learning to formalize practical problems is a mathematical quality that students should have. Students should go through the step-by-step symbolization and formalization process of students' personalized symbolic representation and mathematical representation of specific things. For example, when students have obtained the concepts of rational numbers, similar terms and parallel lines, they need to sum up their definitions in time. What we want is that mathematics should not be divorced from reality and formalized. It is important to find a dynamic balance between grasping the essence of mathematical spirit and formal expression.
In the process of formalizing this mathematical thinking, we can help students explore and discover step by step with the help of the actual operation of learning tools. Hands-on operation is that students realize and reflect their own thinking activities with the help of intuitive activities. Therefore, students must be given enough thinking space. For this reason, a lot of do-do activities are provided in mathematics. The reason why the operation process is needed is because for most mathematical knowledge, it is usually expressed as an algorithm and operation process first, and then as an object and structure, such as the popularization and application process of the commutative law of rational number addition and the associative law of addition. Of course, the operation activities should be appropriate and moderate. When students' intuitive knowledge has accumulated to a certain extent, they must be transformed from intuition to abstraction in time on the basis of enriching appearances.
Second, students' independent exploration involves not only their personal activities, but also the cooperation and communication between classmates.
Knowledge construction is not arbitrary, it has multi-directional sociality and interaction with others, and knowledge construction is in communication; Students' group interactive discussion teaching can promote strategy learning, so cooperative learning is very important for students. In the process of cooperation, students' thinking is divergent. They should not only consider their own ideas, but also compare them with those of their peers to distinguish right from wrong. Students' thinking is constantly advancing or changing, and their own ideas may be improved or denied by their peers, or even replaced, and gradually form a mature solution. In cooperative learning, whether it is to propose solutions, improve solutions or even make mistakes, as long as students actively participate, they will get corresponding experience and improvement. According to different contents, it is an important task of mathematics teaching to properly organize students to carry out activities discussed everywhere in mathematics.
Third, let students really understand mathematics and serve the society with mathematics.
We should not only guide students to upgrade their life experience to mathematical concepts and methods, but also guide students to actively discover, experience and understand mathematics in life and solve practical problems in life with what they have learned. Facing the new mathematical knowledge, actively seek its practical background and explore its application value; In the face of practical problems, we actively try to use the knowledge and methods learned from mathematics to find solutions. Only when students are not limited to the cases provided by teachers and actively look for their own actual background can they find the growing point of knowledge application, further explore its application value and appreciate the value of mathematics. When emphasizing the relationship between mathematics and other disciplines, we should not simply understand this relationship as the calculation of expressions and the measurement of figures in other disciplines, but let students explore the actual background of these expressions and figures in the corresponding disciplines through hands-on operation, induction and thinking. For example, in mathematics, talk about what objects in life are similar in shape to prisms, cylinders, cones and balls. What else can they represent?
Mathematical model is a mathematical structure expressed in a generalized or approximate way by a formal mathematical language, which refers to the characteristics or quantitative relations of things. The key to solving practical problems is to collect the most useful information from practical problems, put forward and find problems from a mathematical point of view, and establish appropriate mathematical models based on these information. Mathematical modeling provides us with an opportunity to connect mathematics with real life. More importantly, students can experience the process of discovering mathematics from the actual situation and get the opportunity to re-create mathematics. Therefore, when solving practical problems, we should avoid formulation, and the focus of teaching is the thinking method in the process of solving problems. Only in this way can students' ability to solve problems be improved. Fifth, mathematics education should not only teach students mathematical knowledge, but also reveal the thinking process of acquiring knowledge. It is the core of cultivating ability to list mathematical thinking methods as basic knowledge of mathematics and develop students' thinking ability. Therefore, the theme of mathematical activities should be basic and important mathematical thinking methods, not simple mathematical facts. Of course, such learning should be carried out through the process of learning activities such as understanding, applying, thinking and expressing specific mathematical knowledge.
To guide students to experience the process of doing mathematics, students should communicate equally and give proper guidance. Teachers should constantly improve the level of listening, asking questions, explaining and actively obtaining information. It is necessary to understand students' real thoughts and take them as the starting point of teaching to provide a good environment for students' learning activities. Always inspire students: How do you know this result? Encourage students to explore, experience from the known, and gain an understanding of new knowledge through their own efforts or cooperation with their peers; When students are faced with difficulties, guide them to find ways to solve problems and summarize the experience gained from the observation of solving problems; When students are not sure about their own mathematical guesses, help them find evidence for their guesses and correct them; When students have doubts about other people's ideas and methods, encourage them to find evidence for their doubts or correct other people's conclusions.
In order to make autonomous exploration a way of learning for students, teachers should always evaluate students: whether they can actively use mathematical knowledge to describe and solve practical problems; Whether you are good at using various methods to solve problems; Whether there is a habit of reflecting on various results; Whether to actively participate in discussion and expression.