[Keywords:] open classroom teaching heuristic subjective mathematics ability
Different historical periods will always advocate some different teaching methods, and some successful teaching models will always give us some enlightenment. A good teaching model should be an open, expanded teaching model that can take its essence and discard its dross and constantly improve with the development of the times.
First, the traditional injection teaching method
In the late 1970s, teachers could only use a blackboard, a textbook and a teaching reference book. In order to have a good public class, teachers usually go to the library to consult relevant books and find some suitable examples and exercises to enrich their classroom teaching. A flexible teaching aid is a small blackboard, which can save writing time and increase capacity. The most wonderful thing is the design of blackboard writing. After a class, blackboard writing is often a knowledge network that echoes before and after, which makes students immediately realize: originally, this class is from shallow to deep, step by step; The complicated problem is to make the conditions more hidden and combine with other knowledge to discover.
[Example 1] This is the beginning of the factorization review open class. Teachers make full use of the idea of transformation and connect the contents of the whole chapter into an organic whole through replacement methods:
Here, the teacher abandons the traditional topic-based training of "formula → practice → next formula → practice ……", but permeates mathematical ideas and methods in problem design, so that students can experience the beauty and profundity of mathematics harmony.
This teaching mode shows the teacher's "teaching"-how to design it? How to express it? At that time, it was in the period of the fifth curriculum textbook reform. 1978,1June, the Ministry of Education issued the Trial Plan of Full-time Ten-year Primary and Secondary Schools (Draft), promulgated a unified national syllabus, and concentrated on compiling the fifth set of national ten-year primary and secondary school textbooks, which began to be used nationwide in the autumn of 1978. This textbook focuses on the choice of basic knowledge, the enlightenment of intelligence and the cultivation of ability. Its main disadvantage is "deep, difficult and heavy", which is not convenient for students to learn and apply by themselves. Therefore, teaching method is the main method of mathematics class in this period, and the advantage of teaching method is simple and efficient. On the one hand, teachers have a profound understanding of knowledge and rich life experience, which can turn abstract into concrete and make profound knowledge easy to understand; On the other hand, this teaching method can give full play to the leading role of teachers, grasp the important and difficult points of teaching, control the teaching rhythm, avoid students taking detours, and quickly realize the teaching purpose, which is very suitable for students who have just stepped out of a special historical period and have a strong desire for knowledge. The focus of the teaching reform of the Ministry of Education is also the adaptation of teaching materials, rather than the change of teachers' behavior. Therefore, with the emergence of education, teaching methods, from word of mouth in primitive society to the present class teaching form, have been continued as a basic method.
Second, pay attention to the heuristic teaching method of cultivating students' ability.
In the 1980s, with the holding of Sino-Japanese summer camps and Sino-American summer camps, the problems of poor hands-on ability and low scores of China students began to be exposed, which also reflected the disadvantages of injection teaching method: the one-way transmission of injection easily made students feel tired, which also led to students' lack of initiative and excessive dependence on teachers, leading to rote learning and mechanical learning; At the same time, the injection teaching method is more for all, and it is not easy to take into account the individual differences of students. It is imperative to reform the injection teaching method that is divorced from students' learning motivation and original knowledge structure.
[Example 2] This is an open class to review the axiom "the shortest line segment between two points".
1. how can ant a get food b the fastest?
(1) When A and B are on the same plane;
(2) When A and B are on the same cube;
(3) When A and B are on the same cuboid (length 5, width 3 and height 4);
(4) When A and B are on the same cylinder;
(5) When A and B (point B is the midpoint of the bus) are on the same cone, it is stipulated that ants should not climb along the bus AB, but along the side.
2. How should bridge PQ be repaired?
(1) Where is the bridge PQ repaired, which can make the sum of the distances from village A and village B shortest?
(2) Where is the bridge PQ repaired to make the sum of the distances between Village A and Village B shortest?
For the question in the above example (1), students can generally answer immediately: "connecting line segment AB". The teacher asked, "Why?" Students can also answer: "The line segment between two points is the shortest!" However, when the teacher showed the question (2), the students' answers were varied, basically walking along the edge of the cube, or adding a diagonal line, still staying in the plane thinking of division, and unable to get the shortest path. At this time, the teacher took out a cube model and unfolded it for the students. Only then did the students think of (1) finding the shortest path. With the teacher's inspiration and help, students break the stereotype and turn the space problem into a plane problem. When students get the question (3), they can imitate the calculation of (2) one after another, but the answer is not unique, so the teacher asks everyone to compare. Students usually choose a direction at will by intuition and expand the side of the cuboid for calculation. Inspired by the teacher, their thinking becomes rigorous: cuboids and cubes are different, with different directions and distances, so we must compare different schemes and choose the shortest path. The following questions, the teacher also let the students think for themselves first. When they are confused, they will convince them in various ways and inspire them to turn the unknown into the known, so as to solve the problem. During this period, teachers began to pay attention to: ① Let students take the initiative to accept meaningful learning. In particular, students should establish the connection between old and new knowledge and "assimilate" new knowledge with old knowledge. ② Implement the heuristic principle in teaching. Pay attention to students' "zone of proximal development" in teaching, guide students to doubt and make them have cognitive impulse.
In the 1980s, junior high school mathematics classroom teaching began to pay attention to the cultivation of ability while paying attention to double basics, so as to combine the characteristics of mathematics subject with students' cognitive characteristics. 1982 《 Mathematics Teaching Syllabus for Full-time Six-year Key Middle Schools 》 clearly points out the relationship between "Double Bases" and "Top Three", and also puts forward clear requirements for learning mathematical thinking methods. In this period, heuristic teaching is the main method of mathematics open class. General classroom design is under the guidance of teachers, so that students can think independently and practice as much as possible. Students usually don't have a serious divergence of "not sending at first", but will solve the problems step by step according to the teacher's design, giving people a sense of fluency and making the listeners clap their hands! The design of classroom questioning shows teachers' skills, which requires teachers to make great efforts to focus on the contact point of knowledge, the occurrence and development of knowledge when preparing lessons, to ensure that the whole process of students' thinking is displayed, so as to facilitate the teaching of the right medicine, so that knowledge is not in a "free state" but incorporated into an optimized cognitive structure.
Third, attach importance to students' subjectivity and dynamically generate classrooms.
Since 1990s, there has been such a paradox in the field of international mathematics education: on the one hand, students in China (including Chinese mainland, Taiwan Province and Hongkong) have achieved excellent results in mathematics learning; On the other hand, western scholars think that China's mathematics learning is "passive acceptance by students" and "repeated practice of standard questions", and the teaching concept is outdated.
Comparison of Mathematics Education between East and West Generally speaking, mathematics education in Chinese mainland and Europe and America is at two extremes. Our task is to find a balance between the two. In order to implement quality education in an all-round way, better solve the curriculum problems left over from the previous seven curriculum reforms, and conform to the trend of the world curriculum reform, the China government has started a wide, comprehensive, in-depth and lasting curriculum system reform.
In the new round of curriculum reform, mathematics classroom teaching increasingly emphasizes "students are the main body of the classroom", which weakens the traces of teachers' design and emphasizes the dynamic generation of the classroom. Teachers are required to respect any thinking sparks of students exploring knowledge, and give them positive affirmation and guidance in time, so that students can truly become the discoverers of knowledge, reproduce the process of knowledge generation, and let them feel the joy of success in thinking and invention.
[Example 3] When learning the nature of isosceles trapezoid, the question is put forward in the form of "doing": "Draw an isosceles trapezoid with parallel lines on a piece of paper and connect two diagonal lines. What are the equal line segments in the diagram? What is an isometric? Is this graph axisymmetric? Try to verify your guess. "
In this paper, please download and install the original text for users who have not installed PDF browser. When explaining that the two base angles on the same base are equal, everyone wondered for a long time. A, the first student who raised his hand to answer, drew Figure (2) and immediately won a piece of approval. The author immediately asked a classmate to explain what he thought A did well. He said, "The advantage of drawing in this way is to turn the unfamiliar isosceles trapezoid problem into a familiar isosceles triangle problem. "Screaming" applause made two students blush with excitement. The author continued to praise: "A student has taken a big step! By adding auxiliary lines, the trapezoidal problem is transformed into a familiar triangular problem, and this is our usual method to solve the trapezoidal problem, which includes an important way of thinking for us to learn mathematics-turning the unknown into the known. In the future, everyone can consider this aspect more when doing trapezoidal questions. "Next, classmate A continues to explain his problem-solving ideas, which can be considered. However, everyone feels that the conditions are not enough, and everyone has doubts and discussions. Immediately, several students raised their hands, and the author asked student B who raised his hand first to answer, "Teacher, I didn't add auxiliary lines like this ..." Student B drew a picture on the blackboard (3), and quickly proved the congruence of two right-angled triangles with HL, thus gradually drawing a conclusion. Next, student C gives Figure (4) and explains the reason. Then the method in Figure (5) was also discovered by the students.
Adding auxiliary lines is a new beginning of trapezoidal learning. It is still a strange and fresh method for students, so it takes more time. Although classmate A didn't find the exact solution in the end, he opened a new way of thinking and made a qualitative breakthrough. Once other students know that there is such a way of thinking, they will continue to find more or even better ways to add threads. As a result of making students think positively, students' learning needs are further met, and teachers' and students' emotions are ignited again and again, gradually rising. When the bell rings, everyone is still wanting more. Throughout the whole class, the teacher has changed from paying attention to the completion of the teaching plan to paying attention to the development and needs of students. When teachers do something in the classroom, they will leave more space for students to do their work. The teacher's "behavior" is more about preparation before class. Only by fully preparing lessons can teachers play an organizational and leadership role in the seemingly inaction classroom. This is exactly what the new curriculum requires teachers to change their ideas. Teachers are good at introducing students' fresh ideas without design into classroom highlights, highlighting students' learning achievements and realizing the harmony between "presupposition" and "generation" in classroom. Only high-quality presupposition can ensure the wonderful and moving generation; Only classroom generation can stimulate the wisdom of thinking. Presupposition makes classroom teaching have rules to follow, and generation makes classroom colorful.
1993 The Central Committee and the State Council issued the Outline of Education Reform and Development in China. Nine-year compulsory education is implemented in primary and junior high schools. 1In June, 1999, the Central the State Council promulgated the Decision on Deepening Education Reform and Promoting Quality Education in an All-round Way. In June, 20001year, the Ministry of Education issued the Outline of Basic Education Curriculum Reform, and put forward the specific objectives of basic education reform: changing the tendency of curriculum to pay too much attention to knowledge transmission, emphasizing the formation of active learning attitude, strengthening the connection between curriculum content and students' life, modern society and scientific and technological development, and paying attention to students' learning interest and experience. We should change the current situation that the curriculum implementation puts too much emphasis on learning and mechanical training, advocate students' active participation, willingness to explore and hard work, and cultivate students' ability to collect and process information, acquire new knowledge, analyze and solve problems, communicate and cooperate.
Fourth, there is no fixed method for teaching, and it is new when teaching.
It can be seen from the current open class that mathematics classroom teaching pays more and more attention to students' subjectivity. In the educational concept, teachers pay attention to students' personal development, so that the so-called "good students", "ordinary students" and "underachievers" can be fully developed and improved on the original basis. Secondly, teachers start with analyzing students' personality factors such as self-awareness, learning style, intelligence or ability, and adopt different educational strategies and methods to teach students in accordance with their aptitude, so as to make everyone's personality develop healthily. Thirdly, we should establish the concept of diversified grades, do more project evaluation and be as many good students as possible. In terms of hardware, teaching AIDS are becoming more and more advanced, and multimedia courseware, physical projection, blackboard writing, etc. can be operated on the touch screen. These advanced teaching tools make students fully feel the convenience brought by the development of science and technology, stimulate their interest in mathematics around them, and enable them to actively participate in mathematics activities. Synchronous transmission system allows more colleagues to attend classes in other classrooms synchronously, which expands exchanges and cooperation and promotes the development of teaching level. As for the advanced and convenient "micro-classroom", its automatic tracking and video recording system can play back the teaching process in detail, facilitate accurate analysis and evaluation of the classroom, facilitate teachers to objectively understand their own teaching situation, and especially promote the rapid growth of young teachers.
In addition to attractive classroom design and teachers' elegant demeanor, there is also an attractive link, which is the evaluation of classes by teachers and experts at all levels. It is an important catalyst for the rapid development of contemporary education, so that teachers' teaching and students' development go hand in hand, and teaching behaviors that violate students' cognitive laws and educational principles will be ruthlessly analyzed and stripped; Teaching methods that can stimulate students' enthusiasm and creativity will be popularized in time. Now teachers are developing faster than ever, and the biggest beneficiaries are students. It can be said that the open class has become a concentrated expression of the teaching philosophy, teaching methods and teaching level of a teacher, a school and a region. It always chooses the best teachers in time, embodies the latest ideas with the latest methods, and becomes the vane of mathematics classroom teaching reform, guiding the majority of mathematics teachers and creating the most vivid classroom with the most effective methods. From the evolution of open classes, we can see the "one spot" of junior high school mathematics classroom teaching-adapting to the development of the times and constantly renewing itself.
References:
[1] Li. Smart classroom comes from the fusion of presupposition and generation [J]. Journal of Middle School Mathematics (Junior High School), 2007, (5).
[2] Li Fenglin. Reflections on the new classroom teaching mode [J]. Mathematics for Primary and Secondary Schools, 2007, (1 1).
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