Function equation thought:

Function thought refers to analyzing, reforming and solving problems with the concept and nature of function. The idea of equation is to start with the quantitative relationship of the problem, transform the conditions in the problem into mathematical models (equations, inequalities or mixed groups of equations and inequalities) with mathematical language, and then solve the problem by solving equations (groups) or inequalities (groups). Sometimes, functions and equations need to be transformed and interrelated to achieve the purpose of solving problems.

Combination of numbers and shapes:

Numbers are intangible, not intuitive, and have many shapes, so it is difficult to be nuanced. The combination of numbers and shapes can make the problem to be studied difficult and simple. Combining algebra with geometry, such as solving geometric problems by algebraic method and solving algebraic problems by geometric method, is the most commonly used method in analytic geometry.

Classification discussion ideas:

When a problem may lead to different results because of different situations of a certain quantity or number, it is necessary to discuss the various situations of this quantity or number in categories.

Thought of equality:

When a problem may be related to an equation, we can solve it by constructing the equation and studying its properties.

General idea:

Starting from the overall nature of the problem, we emphasize the analysis and transformation of the overall structure of the problem, find out the overall structural characteristics of the problem, and be good at treating some formulas or figures as a whole with the "overall" vision, grasping the relationship between them, and carrying out purposeful and conscious overall treatment.

The idea of transformation:

It is through deduction and induction that unknown, unfamiliar and complex problems are transformed into known, familiar and simple problems. The mathematical theories of ancient mathematics, such as trigonometric function, geometric transformation, factorization, analytic geometry, calculus and even ruler drawing, are permeated with the idea of transformation.

Implicit conditional thinking:

Conditions that are not explicitly stated but can be inferred from existing explicit expressions, or conditions that are not explicitly stated but are routines or truths.

Analogical thinking:

Comparing two (or two) different mathematical objects, if they are found to have similarities or similarities in some aspects, it is inferred that they may also have similarities or similarities in other aspects.

Modeling ideas:

In order to describe an actual phenomenon more scientifically, logically, objectively and repeatedly, people use a language that is generally considered rigorous to describe various phenomena. This language is mathematics.

Inductive reasoning thought:

Some objects of a certain kind of things have certain characteristics, and all objects of this kind of things have the inference of these characteristics, or the inference that generalizes general conclusions from individual facts is called inductive reasoning (induction for short). In short, inductive reasoning is from part to whole, from individual to general reasoning.

Extreme ideas:

The idea of limit is the basic idea of calculus, and a series of important concepts in mathematical analysis such as continuity, derivative and definite integral of function are defined by means of limit.