Problem-solving thinking commonly used in high school mathematics competition
According to the characteristics of high school mathematics competition questions, when solving high school mathematics competition questions, the conditions and conclusions given by the questions do not always provide available information directly, which requires the transformation and analysis of the information provided by the conditions and conclusions. At the same time, we are also looking for solutions to problems, and the choice of solutions is the result of the change of thinking mode, so the thinking of solving problems is changeable. The following are four kinds of thinking commonly used in solving problems in high school mathematics competitions:
Native thinking
There are many mathematical problems that show some characteristics as a whole, but it is difficult to find a way of thinking if you think about it as a whole. At this time, you can consider the part of the problem first, "local prompts the whole." We can find a solution to the problem by thinking about the part, or adjust the part of the problem to find the hidden conditions of the problem, and solve the whole problem by solving the part. Because local thinking is simpler than overall consideration, it is often difficult to simplify the problem. When using local thinking strategies, local adjustment and decomposition into parts are often used.
1. Local adjustment
Local adjustment is to constantly adjust all parts of the whole problem by analyzing the similarities and differences between conditions and conclusions, so as to continuously narrow the difference between the initial state and the target state of the problem. On this basis,
Step by step strengthen the requirements and approach the target until the final state of the requirements is reached.
Using adjustment strategies to solve problems should pay attention to the following basic points:
The existing state of the (1) problem is limited;
(2) The goal of adjustment is the final state, and the final state exists, such as seeking the maximum value, and its existence is the premise;
(3) The process of adjustment is to adjust parts to achieve the overall goal.
Break down into several parts
For complex and comprehensive problems, it is often impossible to solve them directly. At this time, we can divide the problem into several parts and solve the whole problem by solving each local problem. In fact, this is also a problem transformation.
Thinking strategy, turn the original problem into several solvable problems. When solving every local problem, we should deal with the relationship between them, which may be independent or progressive, so we need to analyze the problem carefully to ensure the correctness of the thinking direction of solving the problem.
Holistic thinking
The overall thinking strategy is to temporarily avoid the interference of local details or a single factor, grasp the characteristics of the problem as a whole, sort out the problem-solving ideas and find the problem-solving strategy.
Overall strategy is a kind of advanced thinking activity, which has the characteristics of simplicity and jumping, and can improve the speed and accuracy of solving problems. When using the overall strategy, although we observe the characteristics of the problem as a whole and deal with it, we should also pay attention to the relationship between the parts. Using holistic thinking strategy can consider the integrity of the problem or conclusion, and can also grasp the invariance of the whole from the overall characteristics and solve the problem.
For example.
Each congressman has at most three political enemies, which proves that it can be divided into two rooms, so that there is at most one enemy in each congressman's room.
Analysis and solution: when looking at this problem, it seems that every member should consider allocating to qualified ones. The total number of members is not given in the question, which tells us that the above method is not feasible, that is,
Need to adjust the direction of thinking. Re-analyzing the problem, it can be seen from the overall thinking that for the total number of political enemies H in two rooms, when the distribution of satisfying topics appears, H will reach the minimum value, as long as the method of reducing H is studied. At first, each member of parliament is randomly divided into two rooms, and H is the sum of the number of political enemies in his room. Suppose there are at least two political enemies in A's room and at most one political enemy in the other room. Now, if A is moved to another room, the number of H will be reduced by two, but H cannot be reduced all the time. At a certain point, it will reach the minimum value and then reach the expected distribution.
Reverse thinking
Reverse thinking refers to thinking that deviates from the original understanding and explores new development possibilities in the opposite sense. Habit makes it easy for people to form directional thinking when thinking about problems. When they solve problems, they mostly start from conditions and think actively and smoothly with the help of some mathematical thinking methods. However, some topics are difficult or impossible to consider from the front, which requires breaking the mindset, thinking flexibly according to the problems, and adopting reverse thinking strategies. Things are often mutually causal, with the characteristics of two-way and reversibility. When positive thinking is difficult to solve, we can consider turning to negative thinking; When a proposition directly explores and solves difficulties
When, you can go to indirect exploration; When exploring the feasibility of a problem, we can consider its impossibility ... in short, this thinking strategy requires considering the exploration method opposite to the conventional thinking direction and turning to the opposite side of the problem to solve it. In the direction of thinking, it is mainly manifested in difficulties. Straight difficulties are curved, and positive difficulties are reversed.
For example, given irrational numbers A and B, it is proved that at most a group of integer solutions of X and Y satisfy the equation ∣ 3x+ay+1∣+∣ ax-y+3a ∣ = B.
Analysis and solution: First, consider the equation to be solved as a whole, if b0 is like this. The equation given in the problem is an indefinite equation with absolute value.
It is difficult to prove the number of integer solutions of the equation directly from the front, and it is difficult to find ideas. When thinking is blocked, we might as well consider the problem in a different direction. For this problem, we might as well start with the opposite situation. If we can prove that there are no two sets of integers suitable for the equation, then the problem is proved. So you can set two sets of integers to satisfy the equation, and then the sum of absolute values obtained by substituting these two sets of integers into the equation should be equal. After removing the sign of the absolute value, we will find that the two sets of integers are the same, so the problem is solved.
Transform thinking
Transformation is a variable thinking, which means constantly changing the direction of solving problems in the process of solving problems, exploring ways to solve problems from different angles and sides, and grasping the characteristics and problems in solving problems when analyzing and solving problems.
Adjust the thinking of solving problems according to the specific situation [1]. Transformation strategy is the most used thinking strategy in solving problems in high school mathematics competitions, especially in open and research high school mathematics competitions, which have a wide range of knowledge and are generally difficult comprehensive problems. When solving problems, we should not only think deeply, but also think from different angles and directions, otherwise it will be difficult to find a solution. When it is difficult to find an effective solution to the problem directly or directly, it is called transformational thinking to make a breakthrough from the side or the opposite side and turn the problem to be solved into a problem that you are familiar with or can solve.
Problem transformation involves three basic elements: the object, goal and method of problem transformation. The object is the unknown problem we are facing to solve, the goal is the problem we are familiar with or can solve, and the method is the mathematical thinking method. The goal and method of transforming the problem are uncertain, and the transformation angle is different because of the different conditions and conclusions given by the problem. Therefore, when using this thinking strategy to solve problems, there is no fixed mode of thinking transformation, and specific problems should be analyzed. Generally speaking, there are three main links in transforming thinking strategy to deal with problems: (1) changing the known conditions or conclusions of problems; (2) change the form of the problem, such as change.
Volume geometry is plane geometry, and the high dimension is reduced to low dimension; (3) Decomposition and combination, introducing auxiliary elements. Using transformation strategies in solving problems, we can take the following ways to transform problems.
1. Analogical association transformation
There are various relationships between mathematical knowledge, so there must be one connection or another between mathematical exercises. When solving problems, we can grasp these connections according to the specific situation of the problem, and through analogy and
Lenovo explores the train of thought of problem transformation. Observe the conditions and conclusions of the problem to be solved, compare the problem to be solved with the previously solved or familiar problems, and associate it from the aspects of numbers, formulas, graphs with similar characteristics, similar contents and properties, so as to transform the solution of new problems into the solution of old problems that have been mastered, and open up new ideas for solving new problems under the inspiration of old problem solutions.
2. Through decomposition and combinatorial transformation
The structure of some topics in high school mathematics competition is very complex, so it is difficult to find the relationship between the hidden conditions and conclusions in the topics. For this kind of mathematical problems, it is necessary to take the way of decomposition and combination to make the difficult easy, so as to recognize the relationship in the topic. Decomposition and combination are important intellectual activities. For many problems, especially the more difficult ones, it is necessary for us to break them down into some simple or familiar problems that are easy to solve with logical relations according to our own needs, so as to clarify various constraints in the problems to be solved. On the basis of solving the decomposed partial solution, the combination leads to the rearrangement of the relationship structure of the problem to be solved, thus clarifying or solving the original problem.
3. Conversion between general and special
From general to special and from special to general are two interrelated cognitive processes. In solving problems in high school mathematics competitions, the mutual transformation between generality and particularity is a frequently used thinking strategy. Problem-solving thinking has hierarchical characteristics.
Point, in the process of solving the problem, on the basis of understanding the existing knowledge, think more and more deeply until the problem is solved. Therefore, in order to solve the problem, it is sometimes necessary to use the special nature of the problem to draw general conclusions, and sometimes it is necessary to use the general nature to solve special problems.
(1) generalization
When considering a problem, first consider a kind of general problem that contains this problem, which is generalization. As we all know, the generalized problem structure and laws are easier to master. Treat the problem to be solved in a special way.
The transformation strategy of the original problem solution by solving its general form problem is the generalization strategy. The strategy of thinking from the generalization of the problem is an effective way to solve the problem.
(2) specialization
Compared with the general, special problems will be simple, intuitive and concrete, and easy to solve [1]. The solutions of special cases and general cases of problems often have the nature of * * *. In fact, in the process of solving special problems, I often get pregnant.
Solutions to common problems. Therefore, when solving a difficult general math problem, we can solve the special situation of the problem first, and find the way to solve the problem through solving the problem under special circumstances.
4. Intuition and simplification
There are often some abstract concepts or unclear analytical formulas in high school mathematics competition questions, which makes our problem-solving thinking difficult to move forward. In this case, information can be transformed, and some relationships in the problem can be made simple and clear with the help of intuitive things such as graphs or lists, so as to open the way of thinking and find solutions to the problem. The so-called concretization is to turn a more abstract problem into a more concrete or intuitive problem and solve it with the help of intuitive image thinking.
5. Realize transformation by finding auxiliary elements.
The direct solution or proof of some high school mathematics competition problems often can't connect the conditions and conclusions well, and the calculation is very complicated. At this time, it is necessary to find an auxiliary problem or element according to the relationship between conditions and conclusions, thus simplifying complex problems and opening up shortcuts to solve problems.
From the discussion and application of each thinking, students can see that in the process of solving problems in high school mathematics competitions, the thinking strategies of solving problems depend on different topics and different situations, and the direction of thinking always changes according to needs, and different strategies may be adopted and comprehensively applied in solving a problem. This requires students to develop good thinking habits and strengthen the cultivation of problem-solving thinking ability.