Several main viewpoints of high school mathematics

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 a mathematical model with mathematical language, and then solve the problem by solving the equation (group) or inequality (group). Sometimes, functions and equations need to be transformed and interrelated to achieve the purpose of solving problems.

Descartes' equation thought is: practical problem → mathematical problem → algebraic problem → equation problem. The universe is full of equality and inequality. We know that where there are equations, there are equations; Where there is a formula, there is an equation; The evaluation problem is realized by solving equations; The inequality problem is also closely related to the fact that the equation is a close relative.

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).

Step 3 discuss ideas by category

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. Such as solving inequality | a-1| >; 4. It is necessary to discuss the value of A in different categories.

4. Equal thinking

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. Overall thinking

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.

6, classification and integration ideas

(1) classification is a basic logical method in natural science and even social science research.

(2) Select the appropriate classification standard from the specific situation.

(3) Classification is only a means, and classification research is the purpose.

(4)? It is the essential attribute of the idea of classification and integration that there is division and combination, and division before combination.

(5)? This paper studies the classification and integration of mathematical problems with letter parameters, and pays attention to the rigor and thoroughness of students' thinking.

7. Conversion and transformation of ideas

(1) Divide complex problems into simple ones, difficult ones into easy ones and unsolved ones into solved ones.

(2) Flexibility and diversity, there is no unified model, and dynamic thinking is used to find ways and methods to help solve problems.

(3) The college entrance examination pays attention to the commonly used transformation methods: general and special transformation, complex and simple transformation, structural transformation, and equivalent transformation of propositions.

8. Special and general ideas

(1) form an understanding of things through the understanding and research of individual cases.

(2) From shallow to deep, from phenomenon to essence, from part to whole, from practice to theory.

(3) From special to general, and then from general to special.

(4)? Construct special functions and special sequences, find special points, establish special positions, use special values and special equations.

(5)? The college entrance examination is based on new content, and highlighting special and general ideas will become the direction of proposition reform.

9, limited and infinite thoughts:

(1) It is the only way to solve infinite problems by transforming the study of infinity into the study of finiteness.

(2) The accumulated experience in solving infinite problems and transforming finite problems into infinite problems are the direction to solve.

(3) In solid geometry, the surface area and volume of a sphere are solved by division. In fact, it is a typical application of finite and infinite mathematical thought, which divides the sphere into finite times and then sums it up to find the limit.

(4) With the curriculum reform in senior high school, the investigation of new content will be deepened, and the investigation of limited and unlimited will be strengthened.

10, the concept of possibility and inevitability:

Two basic characteristics of (1) random phenomenon, one is the randomness of results, and the other is the stability of frequency.

(2) Find the inevitability in the accident, and then solve the accident with the law of inevitability.

(3) The key points to be examined are the probability of equal possibility events, the probability of one occurrence in mutually exclusive events, the probability of simultaneous occurrence of mutually independent events, independent repeated trials, the distribution list of random events, and mathematical expectations?

1 1, extreme thoughts

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. If you want to ask, "What is the theme of mathematical analysis?" Then it can be summed up as follows: "Mathematical analysis is a subject that studies functions with extreme ideas".

Baidu Encyclopedia-Mathematical Thought