Thinking process of solving math problems in senior one.
The thinking process of solving mathematical problems refers to the whole process of thinking activities from understanding problems, exploring ideas, transforming problems to solving problems and reviewing problems.
For the thinking process of solving mathematical problems, G. Paulia put forward four stages (see Appendix), namely, defining problems, making plans, realizing plans and reviewing. The essence of these four stages of thinking process can be summarized in the following eight words: understanding, transformation, implementation and reflection.
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.
Math problem-solving skills in senior one.
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.
I. Familiarity 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 problem 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, properly constructing auxiliary elements will help to change the form of the topic, communicate the internal relationship between conditions and conclusions (or conditions and problems), and turn unfamiliar topics into familiar ones.
In solving mathematical problems, there are various auxiliary elements of construction, such as constructing graphs (points, lines, planes and bodies), constructing algorithms, constructing polynomials, constructing equations (groups), constructing coordinate systems, constructing sequences, constructing determinants, constructing equivalent propositions, constructing counterexamples, constructing mathematical models and so on.
Second, simplify the strategy.
The so-called simplification strategy means that when we face a complex topic, we try to turn it into one or several simple and easy-to-answer new questions, inspire the thinking of solving the problem through the investigation of the new problems, and control the complexity with simplicity to solve the original problem.
Simplification is the complement and play of familiarity. Generally speaking, we are often familiar with simple problems or easy to be familiar with.
Therefore, in practical problem solving, these two strategies are often combined, but the emphasis is different.
In solving problems, there are many ways to implement simplification strategies, such as finding intermediate links, classified discussion, simplifying known conditions, and properly decomposing conclusions.
1, find intermediate links, and mine implicit conditions:
As far as its background is concerned, some complex comprehensive questions are mostly composed of several relatively simple basic questions, which are properly combined and the intermediate links are removed.
Therefore, it is an important way to simplify complex problems by starting with the causal relationship of problems, finding possible intermediate connections and implied conditions, and decomposing the original problems into a series of interrelated problems.
2, classified investigation and discussion:
In some mathematical problems, the complexity of solving problems mainly lies in that its conditions, conclusions (or problems) contain many difficult-to-identify possible situations. For this kind of problems, it is helpful to simplify complex problems by choosing appropriate classification standards and decomposing the original problems into a group of parallel simple problems.
3. Simplify the known conditions:
Some math problems are abstract and complicated, so it is difficult to start. At this time, we might as well simplify some known conditions in the problem, or even put them aside for the time being and consider a simplified problem first. This simplified question can often answer the original question.
4. Appropriate decomposition conclusion:
Some problems, the main difficulty in solving problems, come from abstract generalization of conclusions, which are difficult to be directly related to conditions. At this time, it is better to guess whether the conclusion can be broken down into several relatively simple parts, so as to solve the original problems one by one.
Third, visualization strategy:
The so-called visualization strategy is that when we face an abstract and elusive topic, we try to turn it into a vivid, intuitive and concrete problem, so as to grasp the relationship between the objects mentioned in the topic with the help of the image of things and find the solution to the original topic.
(a), intuitive chart:
Some math problems are abstract and complicated, which makes it more difficult to understand the meaning. Often due to the abstraction and complexity of the problem, normal thinking is difficult to carry out to the end.
For this kind of topic, using charts or tables to analyze the meaning of the topic is helpful to visualize the abstract content, organize the complex relationship, give a relatively concrete support for thinking, facilitate in-depth thinking and find clues to solve the problem.
(2), intuitive graphics:
Some problems involving quantitative relations are solved by algebraic methods, with rugged roads and a large amount of calculation. At this time, with the help of graphic intuition, we can make a proper geometric analysis of the related quantities in the problem, broaden the thinking of solving the problem and find out a simple and reasonable solution.
(3), intuitive image:
Many problems involving quantitative relations are closely related to the image of functions. Flexible use of image intuition can often control complexity with simplicity and obtain simple and ingenious solutions.
Fourth, the specialization strategy.
The so-called specialization strategy means that when we face a general problem that is difficult to start with, we should take a step back from the general to the special, and first investigate some simple special problems contained in the general situation, so as to broaden the thinking of solving the problem and find the direction or way to solve the original problem from the study of special problems.
Verb (abbreviation of verb) generalization strategy
The so-called generalization strategy is that when we face a special problem with complicated calculation or unclear internal connection, we should try our best to generalize this special problem, find out a method, skill or result that can reveal the general situation of the essential attributes of things, and solve the original problem smoothly.
Sixth, the overall strategy
The so-called integration strategy means that when we face a problem that is difficult to be solved locally or complicated by conventional thinking, we should adjust our perspective in time, take the problem as an organic whole, proceed from the whole, conduct a comprehensive and profound analysis and transformation of the overall structure, and find ways and methods to solve the problem from the study of the overall characteristics.
Seven, indirect strategy
The so-called indirect strategy is that when we face a complicated and difficult problem from the front, or even can't find the basis for solving it on a specific occasion, we should change our thinking direction at any time and think from the opposite side of the conclusion (or problem), so as to solve the original problem more easily.
Four relationships that should be analyzed in solving math problems in senior one.
The Relationship between Examining Questions and Solving Problems
Some candidates do not pay enough attention to the examination of questions, are eager to achieve success, and rush to write, so that they do not fully understand the conditions and requirements of questions. As for how to dig hidden conditions from the problem and stimulate the thinking of solving the problem, it is even more impossible to talk about it, so there are naturally many mistakes in solving the problem. Only by patiently and carefully examining the questions and accurately grasping the key words and quantity in the questions (such as "at least" and "a>0", the range of independent variables, etc. ), and get as much information as possible from it, so as to quickly find the direction to solve the problem.
Second, the relationship between "doing" and "scoring"
To turn your problem-solving strategy into a fractional point, it is mainly expressed in accurate and complete mathematical language, which is often ignored by some candidates. Therefore, there are a lot of "yes but no" and "yes but incomplete" situations on the test paper, and the candidates' own evaluation scores are far from the actual scores. For example, many people lost more than 1/3 points because of "jumping questions" in solid geometry argument, and "substituting proof with pictures" in algebraic argument scored poorly because it was not good at accurately transforming "graphic language" into "written language". Another example is the image transformation of trigonometric function in 17 last year. Many candidates are "confident" but not clear, and the points deducted are not a few points. Only by paying attention to the language expression of the problem-solving process can we grade the "can do" questions.
The relationship between "three quickness" and "accuracy"
The word "quasi" is particularly important in the current situation of large amount of questions and tight time. Only "accuracy" can score, and only "accuracy" can save you the time of examination, while "quickness" is the result of usual training, not a problem that can be solved in the examination room. If you are quick, you will only make mistakes in the end. For example, in last year's application problem No.21,it was not difficult to list piecewise analytic functions, but quite a few candidates miscalculated quadratic functions or even linear functions in a hurry. Although the following part of the problem-solving idea is correct and takes time to calculate, there is almost no score, which is inconsistent with the actual level of candidates. Slow down and be more accurate, and you can get a little more points; On the contrary, if you hurry up and make mistakes, you will not get points if you spend time.
The relationship between four difficult problems and easy problems
After you get the test paper, you should read the whole volume. Generally speaking, you should answer in the order from easy to difficult, from simple to complex. The order of examination questions in recent years is not entirely the order of difficulty. For example, it was more difficult to manage 19 than to manage 20 and 2 1 last year. Therefore, we should arrange the time reasonably when answering questions, and don't fight a "protracted war" on a stuck problem, which will take time and won't get points, and the questions we can do will also be delayed. In recent years, mathematics test questions have changed from "one question to many questions", so the answers to the questions are all set with distinct "steps", which are wide in entrance and easy to start, but difficult to go deep into the final solution, so seemingly easy questions will also have "biting hands" levels, and seemingly difficult questions will also have scores. Therefore, don't be timid when you see the "easy" questions in the exam and the "difficult" questions of new faces. Think calmly and analyze carefully, and you will definitely get the score you deserve.
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