(A) the structure of mathematical thinking
The range of mathematical ideas is very wide, and the basic mathematical ideas commonly used in middle schools are:
The idea of (1) transformation. Mathematics is full of contradictions, complex and simple, difficult and easy, general and special, unknown and known. Through transformation, we can simplify the complex, make the difficult easy, turn the general into the special, turn the unknown into the known, and solve the contradiction. The process of solving mathematical problems is actually the process from conditions to conclusions. Starting from the conditions, we first draw a transitional conclusion, and then gradually transform it into the final conclusion. Therefore, reduction is the most basic idea in mathematics. Specifically, there are transformations such as addition, subtraction, multiplication and division, multiplication and root, exponent and logarithm, high-order to low-order, multivariate to unary, three-dimensional to two-dimensional and so on.
② The idea of functions and equations. Function describes the dependence between quantity and quantity in nature. The idea of function is to abstract the characteristics of quantitative relationship from practical problems and establish functional relationship, so as to study the changing law of variables.
The idea of an equation is to set some unknowns when solving a problem, then find out the equivalent relationship between known numbers and unknowns according to the conditions of the problem, list the equations, and finally solve the problem by solving the values of the unknowns of the equations.
③ The idea of logical division. Also known as the idea of classified discussion, its essence is to determine the classification standard according to the requirements of the problem, classify the research objects, then solve each category separately, and finally draw a comprehensive conclusion.
④ The idea of combining numbers with shapes. The combination of numbers and shapes is to combine quantitative relations with spatial graphics, abstract thinking with image thinking, and to transform quantitative relations into graphic properties and solve algebraic problems with geometric methods, or to transform graphic properties into quantitative relations and solve geometric problems with algebraic methods.
(2) the structure of basic mathematical methods
There are usually two basic mathematical methods:
① Mathematical thinking method. This is a higher-level method in mathematical methods and a thinking method in mathematics, including analysis, synthesis, abstraction, generalization, observation, experiment, association and analogy, conjecture, induction, deduction, generalization and specialization.
② Mathematical problem solving methods. This is a general mathematical problem-solving method, which is universal compared with special problem-solving skills.
Laws include collocation method, method of substitution method, elimination method, substitution method, undetermined coefficient method, parameter method and so on.