This feature can also be expressed by the idiom of "double-base teaching in mathematics". "Double basics" refers to basic knowledge and skills. But "double-base teaching" is not equal to "double-base" itself. As a teaching concept, "double-base teaching" does not simply emphasize laying a foundation, but also includes development on the basis of laying a good foundation. It is a misunderstanding that "double-base teaching" should not be developed.
Mathematics classroom teaching in China has many characteristics different from mainstream research in the world. At one time, these characteristics were either regarded as the object of criticism and sublation, or ignored, while others remained at a simple level and lacked theoretical treatment. We are a little "self-contemptuous" and look down on ourselves. In contrast, we pursue some strange foreign "concepts" and theories that have no practical effect.
Below, we briefly describe six characteristics of Chinese mathematics education, and compare them with relevant foreign references to show the characteristics of Chinese mathematics education.
1. Pay attention to the "import" link.
Tu Rongbao pointed out that China's mathematics teaching is good at deriving new knowledge from old knowledge, and "introducing new lessons" is often the most elaborate part of mathematics teachers' design. Paying attention to "lead-in" is one of the keys to implementing heuristic teaching. A good "lead-in" design will often become the key to the success of a class. After years of accumulation, China has developed into an art of "Introduction to Mathematics".
The "situational setting" introduced from abroad emphasizes the connection with students' daily life, which is just a kind of "introduction". In fact, as far as mathematics classroom is concerned, there are only a few "situations" that can be set up in connection with students' daily life. Most math classes, especially the programmed math content of a large number of "numbers and formulas" operation rules, have no real situation at all. For example, factorization, merging similar terms, power sum exponential operation and so on. It is difficult to set a realistic situation. But can be imported in an appropriate manner. For example, it is feasible to derive factorization from the prime factorization of integers, merge similar items with the naive idea of similar merger, and derive powers from addition as multiplication. There are many unique ways to lead in China's math class, including imagination simulation, suspense setting, story statement, review of old classes, induction of questions, exercise evaluation, bedding, comparative analysis and so on. These import methods are an integral part of heuristic teaching. Recently, it is correct for us to advocate "situational teaching", but it is impossible for people to experience everything directly and get a lot of indirect experiences. Mathematics teaching based on students' daily life situation can only be a kind of reinforcement and supplement of heuristic "guiding people", and can not cancel or replace the setting of "leading-in" teaching link. It is our task to persist in the teaching research of "guiding people to new courses" and make clear the relationship between it and "situational setting"
2. "Trial teaching".
1980s, Gu Lingyuan summarized the excellent cases of mathematics education at that time and put forward the teaching strategy of "guiding as much as possible and giving back the effect", which was popular all over the country. In primary school mathematics education, there is a "trial teaching method" advocated by Qiu Xuehua, which has national influence. They all have the word "try" in their experiences. This is a valuable "creation".
The corresponding concept in the west is "exploration, discovery and creation". But for primary and middle school students, it is difficult to "explore, discover and create" the most basic knowledge that human beings have repeatedly thought and tested in practice for thousands of years in the short nine-year compulsory education.
In mathematics teaching, it is more in line with the reality of basic education to let students "try". The meaning of trying is to put forward your own ideas, which can be right or wrong; You can succeed or fail; You can do it in the end or give up halfway. To try, you don't have to "find out" the results yourself, but you have to have certain ideas, dare to ask questions and dare to experiment. When listening to the teacher's lecture, students "try" against their own right and wrong, and finally grasp the true meaning of knowledge through teacher-student interaction, which is an effective and operable autonomous learning method.
In short, the meaning of "trying to teach" is broader and can be extended to "exploring and discovering". "Trial teaching" can be done in every class, while exploring and discovering mathematical laws can only be done in a small amount. "Try teaching" should be further explored in theory.
3. Teachers' classroom interaction.
The popular foreign "group inquiry", "representative report", "mutual discussion" and "teacher summary" are effective forms of teacher-student interaction, but they are more suitable for small class teaching. If there are more than 30 students in the class and there are many groups, it is difficult for teachers to give comprehensive guidance to the groups.
According to the survey in Cao Yiming, "teacher-class interaction" is the main form of classroom interaction between teachers and students. The class size in China is relatively large, with 40 students in general and 60 students at most. In such a large class, it is very difficult to adopt the teaching methods of group discussion and report exchange. So, how to avoid "full-house irrigation" in mathematics classroom and realize the interaction between teachers and students? In the long-term practice, Chinese mathematics teachers have adopted such measures as "designing questions", "students dictating", "teachers' guidance", "class discussion", "writing on the blackboard", "rigorous expression" and "correcting each other", thus realizing the process of communication and harmonious docking between teachers and students in mathematics language, and finally forming * * * knowledge. This is a creation with China characteristics.
We noticed that when teachers asked math questions, they asked students to stand up and answer them. Students either describe the proof process in oral mathematical language or work out the calculation results by heart. If one student's answer is incomplete, other students will supplement and correct it. Finally, the teacher refines the students' language expressions, forms a rigorous written mathematical language and writes it on the blackboard. In this way, students and students, students and teachers, through the way of "speaking out loud", exposed the mathematical thinking process, carried out mental arithmetic exercises, supplemented and corrected each other in the discussion, the teacher summarized, and finally wrote it on the blackboard in strict written language. This is a harmonious mathematical language docking. The author once received an American colleague, who appreciated it very much.
Small class cooperative learning and large class "teacher-class interaction" have their own advantages and disadvantages. However, the large class size is determined by China's national conditions, and it is still the mainstream.
4. Problem-solving variant exercises.
Variant teaching is adopted in all subjects in our country, but it is more common in mathematics teaching. In particular, the use of variant exercises in solving mathematical problems has become an important feature of Chinese mathematics education. Mathematical variant teaching is a teaching form that changes some connotations of the provided mathematical objects and the presentation forms of mathematical problems from different angles, different sides and different backgrounds, so that the non-essential characteristics of mathematical content are hidden and present, while the essential characteristics remain unchanged. Variant teaching makes students' thinking process have a suitable gradient when doing problems, and gradually increases creative factors; Sometimes a topic can be extended and changed appropriately, providing students with a ladder to try to develop; The combination of exercises should help students summarize all kinds of problem-solving skills, or change problem-solving skills and methods from different angles.
In mathematics problem-solving teaching, variant exercises require teachers to compile training questions arranged in sequence, which provides a ladder for students' thinking development. Although the exercises are repetitive, they are not boring, which is beneficial for students to construct complete and reasonable new knowledge. Each variant has certain innovative significance, but it can lay a solid foundation and realize the teaching concept of "developing on a solid foundation".
A basic law of education is "step by step". In the face of junior students, there used to be a "three small" teaching method of "small slope, small turn and small steps"; A large number of variant exercises at all levels written in the exam guidance book are closely related to the variant exercises of mathematics.
5. Refine the "mathematical thinking method".
Paying attention to the refinement of mathematical thinking methods in mathematics teaching is an important feature of Chinese mathematics education. For a long time, China's mathematics teaching has attached importance to the understanding of concepts, the process of proof and the idea of solving problems, and advocated the teaching of mathematical knowledge generation process. These are all teaching ideas that attach importance to mathematical thinking methods.
From 65438 to 0980, Xu Lizhi formally put forward the theory of "mathematical thinking method" to guide mathematics teaching in primary and secondary schools. This idea quickly gained a warm response in China's mathematics education field and was directly used in classroom teaching. In addition to analyzing general mathematical thinking methods such as synthesis, induction and deduction, associative analogy, etc., we also use methods such as combination of numbers and shapes, reduction method, function idea, equation idea, relation-mapping-inversion principle, geometric transformation, equivalent transformation, gradual approximation, special case anatomy, etc. As for the specific problem-solving methods such as "variable substitution", "undetermined coefficient method" and "cross multiplication", there have always been, and now they are more abundant. The most commendable thing is that these mathematical thinking methods have not stopped at theoretical discussion, but have been put into practice and become the knowledge of every mathematics teacher in China. Mathematics teachers generally have the teaching consciousness of mathematical thinking method, grasp the connotation of mathematical thinking method, solve problems with mathematical thinking method, and summarize and reflect with mathematical thinking method. This is a huge spiritual wealth. When students study mathematics, they will not only solve problems, but also be trained and edified by mathematical thinking methods to develop their own mathematical thinking ability. What a beautiful educational landscape it is!
So far, the western mathematics education field has not put forward the research field of mathematics education which can directly correspond to the "mathematical thinking method", and the formulation of the "process" teaching goal is relatively general.
6. Interpretation of "Practice makes perfect".
"Practice makes perfect" is an integral part of China's cultural tradition and one of the important concepts of Chinese mathematics education. Looking up foreign educational literature, there is no educational theory to support "practice makes perfect". Even though China generally believes that "practice makes perfect", domestic educational literature rarely appears in his works. Education seems to equate "practice makes perfect" with "rote learning". So, why is "practice makes perfect" correct?
Hua, a great mathematician, said in a poem: "A clever plan comes from a clumsy one, and stupidity and wisdom are separated. The wisdom of fools takes a long time to show, and the stupidity of wise men takes a long time to know. Practice first, practice makes perfect. Diligence is a good training, and one point of hard work is one point. " Mr. Chen Shengshen, a master of mathematics, said in a program "Focus Interview": "To do mathematics, you need to be very skilled, do it more, do it repeatedly, and do it for a long time, so that you can understand the mystery and innovate. Inspiration is the result of hard work, otherwise inspiration will not come. " ⑥
This is the case with math, and so is it with math? It is unwise for western educational theory to ignore this point. Mathematics education should take the lead in summing up the law of "Practice makes perfect".
Specifically, practice makes perfect has the following educational connotations: 1. Memory leads to understanding. 2. Speed wins efficiency. 3. The rigorous formation of reason. 4。 Duplicate dependent variant. In addition, the traditional aphorisms such as "Practice makes perfect" and "Reviewing the past and learning the new" have a unique China perspective on the relationship between basic training and innovative thinking.
To sum up, we can make a summary with the three-dimensional diagram ⑦ (see the figure below) of the "Mathematical Double-base Module". First of all, teachers should give full play to the leading role, organize students' trying activities, connect the main piles of basic knowledge into a "mathematical basic knowledge chain", then form a knowledge network through "change", so that practice makes perfect, and then refine mathematical thinking methods to sublimate mathematical ability and form a three-dimensional knowledge module. The mathematical structure of students is formed by the superposition, coupling and connection of "double base" modules.
The elements that appear here are all the characteristics of China.
How to treat "mathematical foundation" is a global problem. The United States launched the "New Mathematics" movement in the 1960 s, emphasizing innovation but ignoring the foundation; So in the 1970 s, he put forward "returning to nature"; /kloc-put forward the slogan of "solving problems" in the 1980 s and advocated innovation and development again; The slogan of 2008 is "Lay a good foundation for success". This is an American "pancake flipping" style.
On the basis of Confucian culture, imperial examination culture and textual research cultural tradition, the "double basics" teaching of mathematics in China is also formed through positive and negative practices. In addition, the characteristics of Chinese mathematics education are not static. "Double base" can be developed. For example, it is feasible to add "basic mathematical activities" and "basic mathematical thinking methods" to become "four foundations". But the "four basics" are developed on the basis of the "two basics". The reform of mathematics education cannot cut off history, abandon tradition and "learn from foreigners".
Precautions:
(1) Tu Rongbao. Knowledge-based teaching environment, edited by Zhang Dianzhou. Double-basic mathematics teaching in China, Shanghai Education Press, 2006, p. 9.
② Experimental Group of Mathematics Teaching Reform in Qingpu County, Shanghai: Learn to Teach, People's Education Press, 199 1.
③ Qiu Xuehua: Try Teaching Method, Fujian Education Press, 1995.
④ Cao Yiming and He Chen: A Study on the Types of Teacher-student Interaction in Junior Middle School Mathematics Classroom, Journal of Mathematics Education, No.5, 2009.