Natural teaching refers to the teaching mode in which teachers guide students to seek the breakthrough point of solving problems with the most fluent and coherent ideas in the teaching of mathematical examples in senior high schools. It is not the teaching of mathematical methods, but the training of students' ability: when faced with unfamiliar mathematical problems, think about what kind of problem-solving strategies to adopt, then quickly determine the direction of problem-solving and develop ideas reasonably. Generally speaking, natural mathematics is to teach students how to think when facing a mathematical problem, which is a decision before applying mathematical methods. Nature here means that education should return to the activities that should belong to the main body of life, to the students' own natural attributes and to their own growth stage, so as to realize the transcendence of education on the basis of nature.
In fact, mathematics is not only the calculation method and various numbers and formulas we learned in school, but also the foundation of harmony, organic nature. Animals, plants, minerals and even raindrops and snowflakes in nature have their own mathematical models or digital forms. According to the message from the editor-in-chief of version A textbook (compulsory) of People's Education Society, the mathematical content appearing in the textbook is the essence and foundation of mathematics after long-term human practice, and the origin and development of mathematical concepts, methods and ideas are natural. If someone thinks a concept is unnatural and imposes it on others, just think about its background, its formation process, its application and its connection with other concepts, and you will find that it is actually a natural product, which is not only reasonable, but even very human.
Nature is an important attribute of students. Before deciding what to teach students, teachers should first understand what students have in their knowledge structure and what their knowledge base is, and on this basis, determine the "nearest development zone" of students. Ignoring the teaching of students' knowledge base is the biggest disrespect for students. Knowledge base reflects the logical starting point of students' learning, and the realistic starting point of students' learning also includes their existing learning experience. In mathematics learning, teachers should be good at analyzing what factors in students' existing knowledge structure and experience system will have a positive impact on the learning of new knowledge, and try their best to mobilize these factors to help students realize the construction of new knowledge. Teaching should not only be unilateral indoctrination, students' mathematics learning is self-construction based on existing knowledge and experience. Only by paying attention to students' internal needs can teaching make them actively engage in learning.
Nature is the essential requirement of mathematics teaching. The subject nature of mathematics and the characteristics of students determine that mathematics teaching is natural. The origin and development of mathematical concepts and mathematical thinking methods are natural, and the process of students' learning and teachers' teaching is also natural. As Paulia said when talking about his problem-solving table: "All the questions and suggestions in our table are natural, simple and obvious, they are just common sense;" But these questions and suggestions are all common words to describe common sense. They suggest some kind of treatment, which is natural for anyone who takes his topic seriously and has some common sense. "
Adapting to the teaching characteristics of nature
In his great teaching theory, Comenius proposed that education should follow the laws of nature. Its main meaning has two aspects: first, education should follow the natural order; Second, education should be based on children's nature. The essence of this natural thought is to study and understand education according to the laws of education itself and from the reality of education, and to deduce its inherent inevitability. Rousseau, known as "Copernicus in education", once again emphasized the principle of natural adaptability in education on the basis of criticizing traditional classical education. He believes that education is the natural growth process of human instinct, and education should abide by the eternal laws of nature, adapt to children's developmental nature, and promote the natural development of children's body and mind.
Diversity. Nature is beautiful because of diversity, and education is more meaningful because of diversity. Since our education is faced with living people, and everyone has a potential thirst for knowledge and self-motivation, then our education should seriously study how to choose the appropriate educational content, educational methods and methods according to each student's different intelligent structure, hobbies and learning methods. Should students adapt to the unchangeable and stereotyped teaching mode based on curriculum standards, or should they actively adapt to each student's personality and needs based on the students' main body? This is the fundamental difference between traditional teaching view and subjective teaching view. Nature teaching emphasizes the need to know each student's personality and what he has learned and what he won't. Only in this way can teaching be targeted and effective.
Make money. In generative teaching, teachers are required to pay constant attention to students' changes and reactions, catch accidental educational opportunities and sparks of wisdom, and make positive responses to students' reactions. In generative teaching, the teacher's influence on students' development exceeds his expectation, and the students will touch the teacher in an unpredictable way, prompting the teacher to take further educational actions instead of completing the prescribed actions. Due to the influence of efficiency mechanism and exam-oriented education, the original interactive teaching process in some schools has become a one-way indoctrination process. In this kind of teaching, the commonly used teaching method is one-way transmission, and the classroom has become a stage for teachers to sing a monologue, and the classroom has become a purgatory for students to learn. Teachers often use control means to try to keep all students' performance under their control. In this case, students seldom really have the desire to develop themselves except remembering the prescribed knowledge of death. Due to the change of teaching thinking mode, generative teaching believes that teaching is no longer a single process of "teaching and receiving", but a process of multi-factor interaction in teaching. Generative teaching exists because of interaction, so interactive methods should be adopted and further improved in the teaching process.
The Realistic Path of Nature Teaching
Mobility: Ordinary is the most natural. In nature teaching, the teacher's task is to guide students to explore ideas to solve problems and make the most common ideas as successful as possible. They want students to feel that solving mathematical problems is not mysterious, and gradually form a positive transfer of thinking mode. In fact, the more common ideas and methods are, the more valuable and vital they are. In traditional classroom teaching methods, teachers are eager to show the solutions to problems after learning concepts and theorems, and pay insufficient attention to the transfer between concepts, theorems and problems to be solved, so that many students unilaterally think that mathematical methods in textbooks are difficult to solve comprehensive problems, so some teachers and students are keen to obtain so-called ingenious problem-solving methods outside textbooks.
The application of transfer teaching can follow the following steps:
Emphasize the direct application of definitions and theorems. Listen to students' ideas, understand the way they deal with relevant information, try to follow students' ideas and compare different solutions repeatedly. Generally speaking, the lower the formula used, the higher the acceptance of the solution.
When analyzing solutions, teachers should always make the following assessments: Are the methods used familiar to students? Are formulas and theorems commonly used? Does the idea of solving problems conform to the thinking habits of most people? Can other solutions to similar problems provide ideological help?
When thinking is blocked, we should strive to "retreat" while "moving forward". Going forward means going further along the existing ideas, or looking for new methods from another angle; Retreat is to go back to the starting point step by step along the existing ideas. Every step back should observe and think about whether the most basic knowledge related to the topic has been ignored. Many times, people's thinking is blocked because people don't pay attention to the simplest knowledge, which is the blind spot of thinking. Consciously starting from the most primitive conclusion is an effective means to reduce blind spots.
Progress: Know what it is and trace it back. Progressive type is based on the development of students' mathematical thinking. Teaching design that conforms to students' thinking characteristics and cognitive rules is not only easy for students to accept, but also will gradually form a good way of thinking with the deepening of learning. In this way, their understanding of mathematics and their attitude towards learning mathematics will gradually change. When they encounter problems that can't be solved, they won't lose their interest in exploration, and they are no longer afraid psychologically.
You can design progressive teaching according to the following steps:
Teachers should be clear about the solutions to the questions, which can be thought of by students and which are not easy to think of. I call the hard-to-think method the core method.
What carrier does the core method appear on? What is the most natural carrier that students feel? List these carriers, and then find out the carriers closely related to this topic. You can also recall what problems students solved when they first came into contact with the core method and why they used this method at that time.
Can the selected vector be transformed into part or all of this topic? If not, what topics are needed for the transition. Then these transition problems are determined, and when the asymptotic relationship between these problems and the problems to be solved is obvious, the preparation work is over.
Put these related topics in the order from easy to difficult. If the topic feels smooth, then the teaching design is completed, otherwise, join the linking topic between the topics with big jumps.
Situational: Thinking is also natural. Situational style means that teachers do not make any preparations before class, and take the difficult questions raised by students as examples to explain them on the spot, so that students can see the teacher's most natural way of solving problems, experience the trajectory of teachers exploring problems, and learn how to think. The characteristic of the situation is that the content in the classroom is unknown to the teacher, while the students are representative problems that have not been solved through hard work.
Situational mode pays attention to the natural formation of mathematical thinking mode, integrates the process of solving problems with thinking mode, concretizes abstract thoughts, and makes the solution of mathematical problems not only rule-based, but also vivid. In short, showing the thinking process can not only let students know the correct way of thinking, but also form the habit of getting lost and prevent the inertia and rigidity of thinking; Showing the thinking process can also enable students to master the thinking method of mathematics learning and understand mathematics knowledge from the height of thinking method.
Success is natural.
The most essential thing in the education system is not restriction, but liberation. Every student has innate creativity, and the creativity of teachers and students is the lasting vitality of educational reform. Paulia believes that ideas should come from students' minds, and teachers should only play the role of "midwives". Therefore, he advocates asking more questions in teaching. He thinks that the most meaningful thing a teacher can give his students is a series of enlightening questions and suggestions. Once students understand this help, they can make the same suggestions to themselves.
Nature teaching is based on the development of students' mathematical thinking, and the teaching design conforms to students' thinking characteristics and cognitive laws, which is not only convenient for students to accept, but also with the deepening of learning, students gradually form a good way of thinking, and their understanding of mathematics and attitude towards learning mathematics will gradually change. When they encounter problems that can't be solved, they won't lose interest in exploration and are not afraid psychologically; When learning new methods, simple mechanical imitation will soon be transformed into conscious and active application; How to analyze and solve mathematical problems will gradually develop correct and flexible thinking habits. The thinking mode and method developed in nature teaching have the strongest ability to transfer. The significance of mathematics teaching is not to let students master much book knowledge, but more importantly, to let students understand the ideological essence of looking at and knowing the world from a mathematical point of view through mathematics learning. Learning to use mathematical thinking helps students develop their intelligence and ability.