In his view, in mathematics teaching, the usual equivalent substitution, combination of numbers and shapes, recursive method and method of substitution can all be called mathematical thinking methods, but they are not the basic ideas of mathematics.
Because when describing these concepts, some specific mathematical contents must be attached, so these concepts are essentially an example, not a general one. In addition, these concepts are not the most basic. For example, regarding equivalent substitution, people can further ask: Why can equivalent substitution be carried out in the calculation process? This means that as a method, equivalent substitution can be derived from other more basic principles. Therefore, it is necessary to establish the principle of judging the basic ideas of mathematics.
We have established two principles:
The first principle, the idea that the emergence and development of mathematics must depend on.
The second principle is the basic thinking characteristics that people who have studied mathematics should have.
According to these two principles, the basic idea of mathematics is summarized into three core elements: abstraction, reasoning and model. The role and relationship of the three in mathematics is roughly as follows:
Through abstraction, people abstract the things related to mathematics in the real world into mathematics, forming the research object of mathematics, and their thinking characteristics are strong abstraction ability;
Through reasoning, starting from the research object of mathematics, under some hypothetical conditions, people can logically get the nature of the research object and the propositions and calculation results that describe the relationship between the research objects, which promotes the development of mathematics. The characteristics of thinking are strong logical reasoning ability;
Through the model, people use the language, symbols and methods created by mathematics to describe the stories in the real world, and build a bridge between mathematics and the real world. Thinking is characterized by strong ability to express the laws of things.
Of course, it is impossible to completely separate the three from the specific mathematical content, especially abstraction and reasoning, abstraction and model.
In the process of reasoning, it is often necessary to abstract concepts and algorithms that are not directly from the real world from the existing mathematical knowledge;
In the process of modeling, it is often necessary to abstract the most essential relationship in the complex realistic background and express it in mathematical language.
On the contrary, abstract processes often need the help of logical reasoning; Judging the relationship between concepts through reasoning, judging what is the independence of propositions and what is the compatibility of propositions, and finally abstracting the axiomatic system;
We found the law in the operation of many cases, verified what is the most essential law through reasoning, and finally expressed the general algorithm with abstract symbols. Therefore, in the process of mathematical research and learning, abstraction, reasoning and model are often between you and me.