1. the idea of transformation and transformation: it is an important basic mathematical idea to turn those problems that are to be solved or difficult to solve into problems that can be solved within the scope of existing knowledge. This transformation should be equivalent transformation, that is, the cause and effect in the process of transformation should be sufficient and necessary, so as to ensure that the result obtained after transformation is still the original result. The learning process of new mathematics knowledge in senior high school is a process of transformation on the basis of existing knowledge and new concepts. Therefore, the idea of transformation is everywhere in mathematics. The application of transformation in problem-solving teaching can be summarized as: turning the unknown into the known, turning the difficult into the easy, and turning the complex into the simple, so as to achieve the purpose of knowledge transfer and problem solving. However, if the transformation is improper, it may also make the problem solving in trouble. example

2. The idea of logical division (that is, the idea of classification and integration): When the essential attributes of mathematical objects are locally different, it is not convenient to solve the problem classified into a single essential attribute, and the appropriate classification criteria are selected according to their different points to solve the problem, and the answer is obtained comprehensively. However, it should be noted that the classification criteria should meet the requirements of mutual exclusion, non-repetition, non-omission and simplicity. The commonly used classification standard in problem-solving teaching is: classification by definition. According to the scope of application of the formula or theorem; Select the appropriate algorithm according to the applicable conditions of the algorithm; According to the nature of the function; According to the change of the position and shape of the graph; According to different situations that may occur in the conclusion, etc. It should be noted that some problems can be solved with the idea of classification, and can be transformed into a new knowledge environment with the idea of reduction or the idea of combining numbers and shapes to avoid classification. Using the idea of classification, the key is to find out the reasons and standards of classification. example

3. The idea of function and equation (that is, the idea of connection or movement change): It is an important basic mathematical idea to analyze and study the quantitative relationship in specific problems with the viewpoint of movement and change, abstract its quantitative characteristics, establish the functional relationship, and use the knowledge of function or equation to solve problems.

4. The idea of combining numbers with shapes: the abstract quantitative relationship in mathematical problems is manifested in the nature (or positional relationship) of some geometric figures; Or abstract the nature (or positional relationship) of geometric figures into appropriate quantitative relations, and combine abstract thinking with image thinking to realize the connection and transformation between abstract quantitative relations and intuitive concrete images, so as to clarify the hidden conditions. This is an important basic mathematical thought to explore the way of thinking in solving problems.

5. Holistic thinking: The focus of dealing with mathematical problems is either on the whole or on the part. It is an important mathematical idea to analyze the structural relationship, corresponding relationship, mutual connection and changing law between conditions and objectives from a global perspective, so as to find the optimal solution. It is the embodiment of the principle of "whole-part-whole" in cybernetics, information theory and system theory in mathematics. In solving problems, in order to facilitate the mastery and application of the overall thinking, what other conditions can this not be used? How to create opportunities to take advantage of unused conditions? ), thinking about the goal (reasoning step by step towards the goal, marking the known and verified with graphics when necessary); Look at the connection, grasp the change, or change; Or number-to-shape conversion, and find the solution. Generally speaking, the larger the overall scope, the better the solution may be.

Under the guidance of holistic thinking, problem-solving skills only need to remember what is known and correctly think about the goals and reasons step by step.

There are some mathematical ideas in middle school mathematics, such as:

The concept of set;

Supplementary thoughts;

Inductive and recursive ideas;

Symmetrical thinking;

Rebellious thoughts;

Analogical thinking;

Parameter variable thought

Finite and infinite thoughts;

Special and general ideas.

Most of them are the concrete embodiment of the basic ideas of mathematics mentioned in this paper in a certain knowledge environment. Therefore, in middle school mathematics, as long as we master the basic knowledge of mathematics, the knowledge points and relations of algebra, trigonometry, solid geometry and analytic geometry, master several commonly used basic mathematical ideas and unify their overall ideas, we will certainly find ways to solve problems and improve our mathematical problem-solving ability.

The Application of the Thought of Conversion in Solving Mathematical Problems

The essence of mathematical activities is the transformation process of thinking. When solving problems, we should constantly change the direction of solving problems, explore ways to solve problems from different angles and different sides, and seek the best method. In the process of transformation, we should follow three principles: 1, familiarity principle, that is, transforming unfamiliar problems into familiar ones; 2. The principle of simplification, that is, turning complex problems into simple ones; 3. The principle of visualization, that is, abstraction is always concrete.

Strategy 1: Forward and reverse transformation

The title and conclusion of the proposition are dialectical unity of causality. When solving a problem, if thinking is blocked from below, we might as well start from the front and think backwards, and there is often another shortcut.

Example: 1: There are * *10 points at the vertex and midpoint of each side of a tetrahedron, among which four points that are not * * * faces are selected, and there are _ _ _ _ _ _ _ ways to take * * faces.

a、 150 B、 147 C、 144 D、 14 1

Analysis: The problem is more complicated from the front. If you consider from the opposite side, it will be much easier to find the total number of four * * * faces first and then use the idea of complementary sets.

Solution: There are many ways to select any four points from 10, among which four of the six points in the face ABC are * * * faces. Similarly, there are many other three faces, and there are also six kinds of * * * faces at the midpoint of each side and the opposite side, and there are three kinds of * * * faces and non * * * faces at the midpoint of each side.

Strategy 2: the transformation from the local to the whole

It is a common way of thinking to analyze problems step by step, but for more complicated mathematical problems, we should grasp things as a whole, don't entangle in details, analyze problems from the system, and don't fight alone.

Example 2: If all sides of a tetrahedron are equal and all four vertices are on the same sphere, then the surface area of this sphere is ().

A, B, C, D,

Analysis: If we use the properties of the circumscribed sphere of a regular tetrahedron to construct a right triangle to solve it, the process is lengthy and easy to make mistakes. But if the regular tetrahedron supplements to form a cube, then the center of the regular tetrahedron and the center of the cube and the spherical surface of the circumscribed sphere are * * * points. Because the side length of the regular tetrahedron is, and the side length of the cube is 1, the radius of the circumscribed sphere is, and (a) should be selected.

Strategy 3: Turn the Unknown into the Known

Also known as analogical transformation, it is an important learning method to cultivate knowledge transfer ability. If we can grasp the known key information in the topic, lock the similarity and skillfully convert the analogy, the answer will come naturally.

Example 3: In arithmetic progression, if, then there is an equation.

(established, analogy to the above properties, in the geometric series, there is the equation _ _ _ _ _ _ _ _.

Analysis: In arithmetic progression, there must be,

,

This is similar to geometric series, because

So it is established.

Logical division thought

For example 1, the set A= is known, and the set B=, if B A, the set of values of the real number a.

Solution A=: It is discussed in two cases.

(1)B=￠, where a = 0；;

(2)B is a unary set, and B=. At this time, it is discussed in two situations:

(i) B={- 1}, then =- 1, a=- 1.

(ii)B={ 1}, then = 1, and a= 1. (secondary classification)

To sum up the above requirements, the collection is.

Example 2. Let the function f (x) = ax-2x+2, and all x values satisfying 1≤x≤4 have the range of f(x)≥ 0 and the real number A. ..

Example 3, known, try to compare the sizes.

analyse

Therefore, we can know that solving this problem must be classified and discussed, and the dividing point is.

Solution:

Summary: The general steps of classified discussion:

(1) Clearly discuss the object and the scope of the object. (i.e. which parameter to discuss);

(2) Determine the classification standard, reasonably divide P, and the standard is unified, with no weight or leakage, and there is no leapfrog discussion. ;

(3) Discuss item by item and achieve phased results. (break the whole into parts and break them one by one);

(4) Summarize and draw a comprehensive conclusion. (The main elements are merged, and the secondary elements are classified and answered).