1 About mathematical methods
At present, there are several views on mathematical methods: ① Mathematical methods are the methods people use when they engage in mathematical activities. ② Mathematical methods not only refer to mathematical research methods (including thinking methods), but also include mathematical learning methods and teaching methods. The so-called "mathematical method" in scientific methodology mainly refers to applying mathematics to solve practical problems.
The so-called method refers to "ways and procedures to solve problems such as thinking, speaking and acting". In short, the method is the way and procedure to solve the problem. There is no doubt that mathematical methods should be a way to solve mathematical problems, or a way to solve mathematical problems. However, this is only one aspect of the extension of the concept of mathematical method. Because solving practical problems with mathematics also requires certain methods and procedures, another aspect of the concept extension of teaching method is the method of solving practical problems with mathematics. The key to solving practical problems by mathematics is to establish corresponding mathematical models for practical problems, so this mathematical method can also be called mathematical model method.
2 on mathematical thought
The concept of mathematical thought is a new concept, and its popularity is only about 10 years. Because of the short time, people's understanding of this concept is still superficial, and even many people just regard it as an "original concept", without really saying what mathematical thought is, but just regard it as a "known".
At present, there are several viewpoints about mathematics thought: ① An excellent mathematics teacher should be good at discovering the "software" part behind the textbook knowledge content-mathematics thought. ② The basic mathematical ideas embodied in primary and secondary school mathematics include "set, relation, mathematical structure, isomorphism, algebraic operation" and so on. ③ Mathematical thought is people's profound understanding of the nature and laws of mathematical scientific research.
Mathematical thought is the existence of mathematics, reflected in people's minds and the result of thinking activities. Obviously, as the result of thinking, mathematical thought has no language to describe. It is completely excavated by mathematicians after serious thinking activities on objective mathematics. Mathematical thought is the sublimation and crystallization of mathematical content and method. What needs to be pointed out in particular is that once the mathematical thought is formed, its significance goes far beyond the mathematical discipline. Mathematical thought is also of guiding significance to other related disciplines.
There are many mathematical ideas that have been recognized and often used, such as the mathematical idea of reduction, that is, the idea of transforming a difficult problem into an easy or solved problem. There are many examples of solving problems with the idea of reduction in mathematics. For example, when the solution of a linear equation group is known, we can change the binary linear equation group into a linear equation group by adding, subtracting or substituting into the elimination method. When the area of a rectangle can be calculated, we can convert the parallelogram into a rectangle with the same product by cutting and filling method, so as to calculate the area of the parallelogram ... The idea of reduction is one of the biggest differences between mathematicians and other scientists. In addition, mathematical ideas such as analysis, synthesis and analogy have long been recognized and applied by everyone.
In addition, mathematical thought has the following educational functions: ① Mathematical thought benefits people for life. A famous mathematician said this when talking about his experience in learning mathematics: "There are many specific teaching knowledge that can be forgotten after learning, but the mathematical ideas expressed by those knowledge will never be forgotten and will be useful to you all your life." As a person in society, in the whole process of receiving teaching and education, he has to learn a lot of mathematical knowledge, not because he really wants to use those hardware knowledge to solve specific mathematical problems in the future, but because they need to absorb the mathematical ideas contained in mathematical knowledge without exception. These mathematical ideas enlighten people in scientific thinking methods, and at the same time cultivate people's scientific attitudes and habits. Clear purpose, clear thinking and accurate behavior are indispensable to people from all walks of life. ② Mathematical thoughts stimulate learners' scientific creative spirit. Every mathematical thought is the crystallization of a shocking intellectual struggle. Its formation process is full of creative thinking of countless people, which marks a leap in the history of inheritance and breakthrough, and embodies a sublimation that stems from practice and is higher than practice. The spirit of scientific creation and the spirit of struggle of creators contained in mathematical thought will certainly stimulate learners' scientific enthusiasm and inspire them to engage in various undertakings with creative spirit. (3) Mathematics thought promotes learners to popularize high-tech technology. Mathematical ideas contained in mathematical knowledge will enable learners to acquire and quickly understand, or comprehend the contents, backgrounds and application prospects of various high-tech, thus gaining the upper hand in the trend of popularization and application of high-tech.
3 the relationship between mathematical methods and mathematical ideas
To sum up, mathematical method and mathematical thought are two completely different concepts, which are both different and related. The difference is that mathematical method is a method to solve mathematical problems, or a method to solve practical problems with mathematics, and mathematical thought is the result of mathematics reflected in people's minds after thinking. Mathematical methods need people to explore, and mathematical ideas need people to explore. Connection is the basis of mathematical method and mathematical thought, and mathematical thought is the deep expression of mathematical method.
The key to the reform of mathematics teaching in middle schools is to attach importance to the teaching of mathematical ideas.
The reform of mathematics teaching in middle schools faces many problems. "What to talk about" is the first question. It can no longer meet the requirements of quality education to talk about the knowledge in books step by step as before. In order to make the middle school mathematics lesson profound, we must attach importance to the teaching of mathematics thought. In order to make students learn profound scientific thinking while learning mathematical knowledge, we must attach importance to the teaching of mathematical thinking, which has been clearly seen from the previous discussion on mathematical thinking. Middle school mathematics teachers fully understand the educational function of mathematics thought, and clarify mathematics thought while explaining and using mathematics knowledge and methods flexibly. Only by attaching importance to teaching ideas can our mathematics teaching enter a higher level. Our students learn not only the knowledge of hardware-mathematics, but also the ideas of software-mathematics, the ability to solve problems, and more broadly, the fundamental idea of how to be a man.