1 Functional thinking

Represent a certain mathematical problem as a function, and use the function to explore the general rules of this problem.

2 The idea of ??combining numbers and shapes

Combine algebra and geometry, for example, use algebraic methods to solve geometric problems, and use geometric methods to solve algebraic problems.

3 Overall thinking?

Integral substitution, superposition and multiplication processing, integral operations, integral element setting, integral processing, complement in geometry, etc. are all integral thinking methods in solving mathematics. specific application in the problem.

4 Transformation of ideas

It consists in transforming unknown, unfamiliar, and complex problems into known, familiar, and simple problems through deduction and induction.

5 Analogical thinking?

Compare two (or two types of) different mathematical objects. If you find that they are the same or similar in some aspects, then infer them There may be similarities or similarities in other aspects. ?

Extended information:

Function thinking refers to using the concept and properties of functions to analyze, transform and solve problems. Equation thinking starts from the quantitative relationship of the problem, uses mathematical language to transform the conditions in the problem into a mathematical model (equation, inequality, or a mixed group of equations and inequalities), and then solves the problem by solving the equation (group) or inequality (group) to solve the problem. Sometimes, functions and equations are also transformed and connected to each other to achieve the purpose of solving problems.

Descartes’ idea of ??equations is: practical problems → mathematical problems → algebraic problems → equation problems. The universe is full of equations and inequalities. We know that wherever there are equations, there are equations; wherever there are formulas, there are equations; evaluation problems are realized by solving equations...and so on; inequality problems are also close relatives and closely related to equations. Listing equations, solving equations, and studying the properties of equations are all important considerations when applying equation thinking.

Function describes the relationship between quantities in nature. Function thinking establishes a functional relationship-type mathematical model for research by proposing the mathematical characteristics of the problem.

It embodies the dialectical materialist view of "connection and change". Generally speaking, the idea of ??a function is to construct a function and use the properties of the function to solve problems. The properties often used are: f(x), the monotonicity, parity, periodicity, maximum and minimum values, image transformation, etc. of f(x) , we are required to master the specific characteristics of linear functions, quadratic functions, power functions, exponential functions, logarithmic functions, and trigonometric functions.

In solving problems, being good at exploring the implicit conditions in the problem, constructing analytical expressions of functions and using the properties of functions is the key to applying function ideas. Only when the given problem is observed, analyzed, and judged more deeply, fully, and comprehensively can the connection between one and that be made and the function prototype constructed. In addition, equation problems, inequality problems and certain algebra problems can also be transformed into related function problems, that is, using functional ideas to solve non-function problems.

Function knowledge involves many knowledge points and is wide-ranging. It has certain requirements in terms of concept, application and understanding, so it is the focus of the college entrance examination.

Several common question types where we apply functional ideas are: when encountering variables, constructing function relationships to solve problems; related inequalities, equations, minimum values ??and maximum values, etc., are analyzed using the function perspective ; In mathematical problems containing multiple variables, select the appropriate main variable to reveal the functional relationship.

Translate practical application problems into mathematical language, establish mathematical models and functional expressions, and apply knowledge of functional properties or inequalities to answer them; in arithmetic and geometric sequences, general formulas and sums of the first n terms Formulas can be regarded as functions of n, and sequence problems can also be solved using functional methods.

The main reasons for the discussion of classification are the following aspects:

① The mathematical concepts involved in the problem are defined by classification. For example, the definition of |a| is divided into three situations: a>0, a=0, and a<0. This type of classification discussion question can be called conceptual type.

② The mathematical theorems, formulas, operational properties, and rules involved in the question have scope or conditional restrictions, or are given in categories. For example, the formula for the sum of the first n terms of a geometric sequence can be divided into two situations: q=1 and q≠1. This type of classification discussion question can be called a qualitative type.

③ When solving questions containing parameters, discussions must be based on the different value ranges of the parameters. For example, when solving the inequality ax>2, we can discuss three situations: a>0, a=0 and a<0. This is called a parametric type.

In addition, certain uncertain quantities, uncertain shapes or positions of figures, uncertain conclusions, etc., are mainly discussed through classification to ensure their completeness and make them deterministic.

When conducting classification discussions, the principles we must follow are: the objects of classification are determined, the standards are unified, no omissions, no duplications, scientific divisions, prioritization, and no skipping discussions. The most important one is "nothing missing, nothing heavy".

When answering classification discussion questions, our basic methods and steps are: first, determine the scope of the discussion object and all objects discussed; secondly, determine the classification standards, and correctly carry out reasonable classification, that is, the standards are unified and different. The omissions are not repeated and the classifications are mutually exclusive (no duplication); then the classifications are discussed step by step and graded to obtain phased results; finally, a summary is made to draw a comprehensive conclusion.

Reference: Baidu Encyclopedia-Mathematical Thinking Methods