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.

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.

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.