The research of "new basic education" puts forward that "the unique value of subjects for students' development should be realized through teaching", which is noble and realistic. The lofty goal is to let students finally establish a unique way of thinking. In fact, this way of thinking needs to be established through teaching infiltration in every class every day. So, is this goal possible?
At the beginning of the research, we found it almost impossible to achieve! Because in the classroom, teachers are often limited to the reform of teaching forms. Taking the calculation teaching of primary school mathematics as an example, due to the different calculation forms and results in the textbooks at that time, the whole calculation knowledge was divided into different types, such as oral calculation and written calculation, sequential operation and simple operation according to the law, accurate operation and estimation. Textbooks are also arranged in the form of knowledge points and examples according to the difficulty of knowledge. In teaching, teachers follow the style of teaching materials and explain a knowledge point and an example in isolation. Although there are many forms of students' "active" activities in class, such as questioning, questioning and group discussion, we can find that the depth of students' thinking is "passive" response and obedience: when teachers teach simple calculation methods, students will not have estimation methods; When teachers teach estimation methods, students will not use simple calculation methods. In this way, according to the teaching method of one knowledge point and one book knowledge example, students will also "cooperate" with teachers, and they will question, discuss and think about various methods around knowledge points. The essence of this teaching form is to "teach" students to master the calculation method mechanically, and "educate" the basic lifestyle is passive adaptation. In order to make students' thinking really active, "education" takes active development as the basic way of life, and "new basic education" realizes that it is necessary to take the internal structure of mathematical knowledge as educational resources and establish a holistic view of mathematics teaching. Because structure has much stronger organization and migration ability than knowledge points, it not only enables students to firmly grasp and skillfully use the knowledge related to structure and internalize it, but more importantly, through the study of structure, students can be happy and positive in guessing and analogy because of the support of structure, which can urge students to really take the initiative to invest in their thinking and form their active learning attitude and ability. On this basis, students can be further equipped with the methods of discovering and forming structures and the ability to master and use structures flexibly. As for the teaching of structure, we adopt the teaching strategy of "two stages in distance": firstly, we need to reorganize the existing teaching contents according to the internal logic of mathematical knowledge to form a structural chain; Secondly, teachers need to break the original teaching method of a knowledge point and an example of "uniform movement" and divide the learning of each structural unit into "teaching structure" stage and "using structure" stage. In the stage of "teaching structure", inductive discovery is mainly adopted, so that students can fully experience discovery and construction from practical problems and gradually form knowledge structure, learning method and step structure. The teaching time at this stage can be moderately slowed down. In the stage of "using structure", students are mainly required to make active guesses, analogies and verifications by using structure. Because students have been able to master and flexibly use this structure for active learning, the teaching time at this stage can be accelerated.
Taking the teaching of addition, subtraction, multiplication and division in primary school mathematics as an example, teachers should establish the overall consciousness of integrating oral calculation, written calculation, simple calculation and estimation, take the operation structure of teaching written calculation as the main line, and infiltrate other calculation methods into it. When teaching the operation structure of addition pen calculation, "teaching structure" is the main way; When teaching the written structure of other methods, the way of "using structure" is the main way. What we expect to achieve is not only the students' mastery and flexible application of the operation structure, but also the improvement of teachers' overall awareness of mathematics teaching, efforts to create conditions, opportunities for students' various activities, learning to capture the resources generated by students as an opportunity, and comprehensively infiltrating oral calculation, simple calculation, estimation and other methods into teaching, so as to cultivate students' awareness and ability to judge quickly and choose methods flexibly. We believe that, first of all, computing teaching, which combines various computing methods, is the carrier, which serves to cultivate students' ability to judge and choose flexibly and to cultivate students' overall ability to grasp problems. Secondly, the teaching of calculation, which combines various calculation methods, provides a stage and development space for students to make judgments according to specific conditions and conditions, and makes it possible for students to learn and use various calculation methods flexibly. In this way, students' thinking can be developed positively, and the knowledge goal of computing teaching can be realized naturally.
2. Take the process of creation and development of mathematical knowledge as educational resources.
In the past, mathematics teaching paid more attention to the memory and application of mathematical knowledge, emphasizing deduction and neglecting induction. Students only know symbols by memory and are tired of imitation and practice, but they don't know the ins and outs of knowledge. Taking the process of creation and development of mathematical knowledge as an educational resource can not only help students understand the ins and outs of mathematical knowledge, but also make them experience the process of creation and development of mathematical knowledge, feel the basic ideas and methods of mathematics, feel the abstraction and strength of mathematics, form the internal driving force for learning mathematics, and gradually establish a unique way of thinking, which is irreplaceable by other disciplines and has unique educational value only in mathematics. In order to restore the original appearance of the production and development of mathematical knowledge, it is necessary to reorganize and process it according to the process of knowledge discovery and development of teaching materials, and realize the communication between book knowledge and mathematical knowledge.
For example, in the teaching of "congruent triangles's Judgment Theorem" in mathematical geometry in middle schools, traditional textbooks do not describe the process of people discovering judgment theorems, but arrange the discovered results (four judgment theorems) in the form of teaching one theorem and one example in one class, and present them to students in a deductive way. This way of presentation, first of all, is easy to cause students to memorize and copy mechanically; Secondly, it is easy for students to exist in order to learn these judgment theorems; More importantly, it is easy to lead to the depression and passivity of students' thinking. Because it lacks attention to students' learning needs; Lack of attention to how students experience and experience the discovery process of congruent triangles's judgment theorem; Lack of attention to how students learn meaningfully. That is, there is a lack of thinking and research on the organic process and value of students' learning congruent triangles's judgment theorem.
In order to reveal the true face of the discovery and development of congruent triangles's judgment theorem, we first analyze the students' existing learning experience, as well as the puzzles that often appear in learning and the premise problems that need to be solved. How many conditions do you need to determine a triangle? Congruent triangles's judgment theorem requires at least several conditions. According to the three conditions, how many combinations of the sides and angles of a triangle are there? In many combinations (there are six kinds of * * *), can they all be decision theorems? Wait a minute. Then, the content of the textbook was reorganized and processed according to the process of its discovery and development: in the first teaching, students were focused on understanding the ins and outs of congruent triangles's judgment theorem from the overall perception, experiencing mathematical activities such as observation, discovery, conjecture, verification, induction and generalization, experiencing the formation process of congruent triangles's judgment theorem, feeling the mathematical ideas and methods infiltrated into it, and feeling the way of thinking from accident to necessity and from special to general. In the second or third teaching, students are emphasized to quickly judge the judgment conditions and flexibly choose the judgment theorem, and master the writing format when using the judgment theorem to prove. The teaching design of the first teaching adopts the method of inductive discovery to teach. First of all, the premise of judging the congruence of triangles is put forward to stimulate students' learning needs and thirst for knowledge; Then, in the form of cooperation between two people, choose one or two of the six combinations for guessing and experimental verification; Then the whole class exchanges and summarizes, and comes to the conclusion that four of the six combinations can be decision theorems. Here, two of them do not constitute a combination of judgment theorems, which will become an important resource for students to form a correct understanding.
If the teaching of congruent triangles's judgment theorem is put into the knowledge structure of geometry judgment theorem in middle school, this teaching method can be applied, and the teaching strategy of "long distance and two stages" can be adopted. The teaching structure is the main part of the judgment theorem in middle school geometry at the initial stage, and the study of the judgment theorem in the later stage can make students actively think, guess and discover by using the structure. We believe that the value of this kind of teaching for students' development lies in: not only let students perceive and understand the context of the judgment theorem as a whole and form a meaningful understanding, but also let students experience and appreciate the formation process of the judgment theorem and feel the ideas and methods of mathematics. More importantly, students have mastered the knowledge structure and learning method structure of the judgment theorem, and when they learn the judgment theorem in the future, they will have the possibility of active guessing and analogy, which is very important for students to think actively and form an active learning mentality. In the experiment, we not only take the whole and internal knowledge structure of mathematics and the process of creation and development of mathematical knowledge as educational resources from the perspective of knowledge, but also take the people and history invented and created by mathematics as educational resources from the perspective of people, and take the basis and life experience of students learning mathematics as educational resources.
3. Take people and history invented by mathematics as educational resources.
In the long history of human mathematics development, the brilliant Starlight Glimmer. Far, can be traced back to the discovery of Pi Zu Chongzhi; Near, can be associated with Sue, Chen Jingrun. Many mathematical inventions or creations at home and abroad, large and small, fully reflect the wisdom of predecessors. Although it is mentioned in traditional mathematics textbooks, it is mostly just an introduction, which is presented by future generations through memory or using the achievements of predecessors, which leads to these important educational resources becoming forgotten corners. Mathematics teaching needs to be deeply developed, so as to realize the communication between book knowledge and mathematical inventions and history, show the most intelligent part of mathematical inventions as a rich resource to realize the value of mathematics education, and let students feel wisdom, practice wisdom and embody wisdom in the process of "re-creation" of these mathematical inventions.
For example, in the teaching of calculating the circumference, the previous teaching focus was to calculate the circumference with the pi discovered by Zu Chongzhi. In order to let students experience the rediscovery of pi in Zu Chongzhi, the experimental teacher provided students with many disks of different sizes to study the relationship between pi and radius and diameter. After research, students have made many discoveries: some students found that the circumference is a little more than six times the radius and a little more than three times the diameter; Some students found that the radius is 0. 16 times of the circumference, and the diameter is 0.3 times of the circumference. Some students found that the circumference of a circle is a little more than twice the sum of radius and diameter; By analogy, on this basis, teachers guide students to analyze, compare, summarize and generalize. Finally, these numerous findings are summed up in one point: the circumference of a circle is 3. 14 times the diameter. In this class, students feel, practice and reproduce Zu Chongzhi's wisdom. Teachers are surprised at the potential of students, at the discovery of students, and feel the inner dignity and joy of teachers' profession!
For another example, in the past, the teaching of centimeter cognition and angle measurement focused on how to measure with straightedge and protractor, while ignoring the value of the invention and creation process of straightedge and protractor. These inventions are the crystallization of predecessors' wisdom. If they are developed into rich educational resources, students will become smarter in the process of "re-invention".
4. Take students' learning foundation and life experience as educational resources.
If the establishment of the overall structure view of mathematics teaching needs to be recognized and advocated, then the connection between book knowledge and people's life world and children's experience world has been recognized and vigorously practiced by teachers. However, as far as I can see, more and more cases are carried out with the help of "additive thinking". That is to say, using "mathematical problem+life situation" to realize the connection, thinking that as long as "life situation" is set in the classroom (sometimes it doesn't exist in life), it will be connected with the life world, ignoring the real meaning of book knowledge in daily life and the process experience of abstracting mathematical problems from life situation, so "communication" often seems superficial and far-fetched. For example, in the teaching of solving the application of right triangle in middle school mathematics, the teaching goal set by a teacher for the teaching content is to further improve students' ability to solve practical problems by using mathematical knowledge through teaching. In order to achieve this teaching goal, teachers combined with "real life" and created the following problem situations:
There are two buildings in a residential area, from point A to point E on building A.
A 30-meter-long propaganda banner was hung on the roof of building B at point D.
The elevation of point A at the top of the banner is 30, and the elevation of point E at the bottom of the banner is 30.
The attachment angle is 20. Find the horizontal distance BC between two buildings A and B.
(accurate to 0. 1 m). The teacher's expected answer is to find the horizontal distance BC between two buildings by solving the right triangle. Students know what it means, and cooperate with the teacher very much, drawing pictures, adding straight lines to construct a right triangle, calculating the relationship between the sides and angles of the right triangle, and finally getting the answer consistent with the teacher's expectation.
We know that the basic principle of using mathematical knowledge to solve practical problems is to simplify the complex and turn the difficult into the easy. Turn the hidden into the obvious. Here, turning the hidden into the obvious refers to revealing and revealing mathematical problems or mathematical models hidden in daily life situations. Finding the distance between two buildings is indeed a practical problem in life, but to solve the above problems, we can completely use the method of estimation or direct measurement, and we don't need to use the redundant transformation of elevation angle and attached angle between a point at the top of one building and another building to calculate the distance between two buildings. Obviously, the "creation" of the teacher's "situation" is at least blunt or incomplete. But it is worth reflecting on: Why do the whole class solve this simple practical problem by solving the right triangle in accordance with the idea of the problem situation designed by the teacher? Why didn't a student question the teacher's design? From this, we can at least see that the combination of mathematical knowledge and real life must be based on truth and possibility, otherwise it will be fundamentally divorced from practice. On the other hand, the teaching method of "what to teach" and "what to practice" around knowledge points for a long time has made students' thinking form a passive obedience mode, and they are often used to thinking about the method of solving problems according to knowledge points, which is manifested in solving problems for the sake of solving problems and rarely thinking about the true meaning of problems. This is a state that we should pay great attention to avoid and change in the reform.