1. Functional concept:
A mathematical problem is expressed by function, and the general law of this problem is explored by function. This is the most basic and commonly used mathematical method.
2. The combination of numbers and shapes:
"Numbers are invisible, not intuitive, and numerous shapes make it difficult to be nuanced", and the application of "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. For example, find the root number ((A- 1)2+(B- 1)2)+ root number (A 2+(B- 1)2)+ root number ((A- 1) 2+B).
3. Discuss ideas by category:
When a problem may lead to different results because of different situations of a certain quantity, it is necessary to classify and discuss the various situations of this quantity. Such as solving inequality | a-1| >; 4. It is necessary to discuss the value of A.
4. Equation concept:
When a problem may be related to an equation, we can solve it by constructing the equation and studying its properties. For example, when proving Cauchy inequality, Cauchy inequality can be transformed into a discriminant of quadratic equation.
5. 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 holistic thinking method is widely used in simplification and evaluation of algebraic expressions, solving equations (groups), geometric proof and so on. Integral substitution, superposition multiplication, integral operation, integral demonstration, integral processing and geometric complement are all concrete applications of integral thinking method in solving mathematical problems.
6. Change ideas:
It is through deduction and induction that unknown, unfamiliar and complex problems are transformed into known, familiar and simple problems. Mathematical theories such as trigonometric function, geometric transformation, factorization, analytic geometry, calculus, and even rulers and rulers of ancient mathematics are permeated with the idea of transformation. Common transformation methods include: general special transformation, equivalent transformation, complex and simple transformation, number-shape transformation, structural transformation, association transformation, analogy transformation and so on.
7. 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.
8. Analogy:
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.
9. Modeling ideas:
In order to describe an actual phenomenon more scientifically, logically, objectively and repeatedly, people use a language that is generally regarded as strict to describe various phenomena, which is mathematics. What is described in mathematical language is called a mathematical model. Sometimes we need to do some experiments, but these experiments often use abstract mathematical models as substitutes for actual objects and carry out corresponding experiments. The experiment itself is also a theoretical substitute for the actual operation.
10. Back to thinking:
The idea of transformation is to turn the unknown into the known, the complex into the simple and the difficult into the easy. For example, fractional equations are transformed into integral equations, algebraic problems are transformed into geometric problems, and quadrilateral problems are transformed into triangular problems. The methods to realize this transformation are: undetermined coefficient method, collocation method, whole generation method and the transformation idea of turning dynamic into static and abstract into concrete.
1 1. 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.
In addition, there are mathematical ideas such as probability statistics, for example, probability statistics refers to solving some practical problems through probability statistics, such as the winning rate of lottery tickets, the comprehensive analysis of an exam and so on. In addition, some area problems can be solved by probability method.