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The first stage: understanding the problem is the beginning of problem-solving thinking activities.
The second stage: transformation is the core of problem-solving thinking activities, an active attempt to explore the direction and way of problem-solving, and a process of selecting and adjusting thinking strategies.
The third stage: plan implementation is the realization of problem-solving process, which includes a series of flexible application of basic knowledge and skills and concrete expression of thinking process, and is an important part of problem-solving thinking activities.
The fourth stage: the problem of reflection is often ignored by people. It is an important aspect of developing mathematical thinking, the end of a thinking activity process and the beginning of another new thinking activity process.
Skills of solving mathematical problems
In order to make the direction of memory, association and conjecture clearer, the thinking more vivid and further improve the effectiveness of inquiry, we must master some problem-solving strategies.
The basic starting point of all problem-solving strategies is "transformation", that is, the problem that is faced is transformed into one or several new questions that are easy to answer, so that the idea of solving the original problem can be found through the investigation of the new problems, and finally the purpose of solving the original problem can be achieved.
Based on this understanding, the commonly used problem-solving strategies are: familiarity, simplification, intuition, specialization, generalization, synthesis and indirection.
First, be familiar with the strategy.
The so-called familiarity strategy means that when we are faced with a strange topic that we have never touched before, we should try to turn it into a previously solved or familiar topic, so as to make full use of the existing knowledge, experience or problem-solving mode and solve the original topic smoothly.
Generally speaking, the familiarity with the topic depends on the knowledge and understanding of the structure of the topic itself. Structurally, any solution contains two aspects: conditions and conclusions (or problems). Therefore, if you want to turn unfamiliar problems into familiar ones, you can make more efforts in changing conditions, conclusions (or problems) and their contact information.
Commonly used ways are:
(1) Fully associate and recall basic knowledge and questions:
According to Paulia's point of view, before solving a problem, we should fully associate and recall the same or similar knowledge points and problems as the original problem, and make full use of the ways, methods and conclusions in similar problems to solve the existing problems.
(2) Analyze the meaning of the problem from all directions and angles:
For the same math problem, we can often understand it from different sides and angles. Therefore, according to one's own knowledge and experience, adjusting the perspective of analyzing problems in time is helpful to better grasp the meaning of problems and find the familiar direction of solving problems.
(3) Appropriate construction of auxiliary elements:
In mathematics, the topic of the same material can often have different expressions; Conditions and conclusions (or problems) are also related in many ways. Therefore, the auxiliary element is appropriately constructed,