Solving mathematics problems in senior one is an important means to deepen knowledge, develop intelligence and improve ability. Standardized problem solving can develop good study habits and improve thinking level. A certain amount of practice is necessary in the learning process, but not as much as possible. The tactics of asking questions about the sea can only increase the burden on students and weaken the role of solving problems. In order to overcome the tactics of questioning the sea and strengthen the role of solving problems, we must strengthen the standardization of solving problems.
Problem-solving norms include four aspects: examination norms, language expression norms, answer norms and reflection after solving problems.
First, the examination standards
Examining questions is the key to solving problems correctly, and it is the process of analyzing, synthesizing and seeking ideas and methods for solving problems. The examination of questions includes three parts: defining conditions and objectives, analyzing the relationship between conditions and objectives, and determining the ideas and methods of solving problems.
The analysis of (1) condition is to find out the known conditions explicitly told in the topic, and to find out the implied conditions in the topic and reveal them.
Target analysis is mainly to clarify what is needed or what needs to be proved; Turn complex goals into simple goals; Turn abstract goals into concrete goals; Turn an unattainable goal into an achievable goal.
(2) Analyze the relationship between conditions and objectives. Every mathematical problem consists of several conditions and objectives. The problem solver needs to find out what is missing from the condition to the goal on the basis of reading the topic. Either deduce from the conditions, analyze the objectives, or draw a related sketch, mark the conditions and objectives on the diagram, and find out their internal relations, so as to successfully achieve the purpose of solving the problem.
(3) Determine the way to solve the problem. There are a series of inevitable connections between the conditions and goals of a topic, which is a bridge from conditions to goals. Which connections to use to solve problems needs to be decided according to the mathematical principles followed by these connections. The essence of solving problems is to analyze which mathematical principle these connections conform to. Some topics, this connection is very hidden, and it must be carefully analyzed before it can be revealed; Some problems have multiple matching relationships, which is why a problem has multiple solutions.
Second, the language description standard
Language narration (including mathematical language) is a process of expressing problem-solving procedures and an important link in solving mathematical problems. Therefore, language narration must be standardized. Standardized language narrative should be clear, correct, complete, detailed and appropriate, and the words should be well-founded. Mathematics itself has a set of standardized language system, so we must not make up mathematical symbols and mathematical terms at will, which makes people confused.
Third, the answer specification
Answer specification means that the answer is accurate, concise and comprehensive, and attention should be paid not only to the verification and selection of the result, but also to the integrity of the answer. To achieve standardized answers, it is necessary to examine the objectives of the questions and answer them according to the objectives.
Fourth, reflection after solving the problem.
Reflection after solving a problem refers to retrospective reflection on the process, method and knowledge used to solve the problem after solving the problem. Only in this way can we effectively deepen our understanding of knowledge and improve our thinking ability.
Thoughts on solving problems in junior high school mathematics I. the solution of multiple-choice questions
1, direct method: according to the setting conditions of multiple-choice questions, the requirements of the questions are finally obtained through calculation, reasoning or judgment.
2. Special value method: (Special value elimination method) Some multiple-choice questions involve mathematical propositions related to the range of letters. When solving this kind of multiple-choice questions, we can consider selecting some special values from the range of values, substituting them into the original proposition for verification, and then eliminating the wrong ones and keeping the correct ones.
3. Exclusion method: Return the four conclusions given in the topic to the original topic one by one for verification, and eliminate the mistakes until the correct answer is found.
4. Phase-out method: If you don't do it step by step in the calculation or derivation process, but step by step, use both? Take a walk and see? Every step of the strategy is compared with four conclusions, and the impossible is ruled out, so that three wrong conclusions may be ruled out before the last step.
5. Number-shape combination method: According to the internal relationship between the conditions and conclusions of mathematical problems, it not only analyzes its algebraic significance, but also reveals its geometric significance, so as to skillfully and harmoniously combine the quantitative relationship with the graphics, and make full use of this combination to seek ideas and solve problems.
Second, the commonly used mathematical thinking methods
1, the idea of combining numbers and shapes: According to the internal relationship between the conditions and conclusions of mathematical problems, we not only analyze their algebraic significance, but also reveal their geometric significance, so as to skillfully and harmoniously combine the quantitative relationship with figures, and make full use of this combination to seek the idea of disintegration and solve problems.
2. The idea of connection and transformation: Things are interrelated, restricted and transformed. All parts of mathematics are also interrelated and can be transformed into each other. When solving problems, if we can properly handle the mutual transformation between them, we can often turn the difficult into the easy and simplify the complicated. Such as: substitution transformation, known and unknown transformation, special and general transformation, concrete and abstract transformation, partial and whole transformation, dynamic and static transformation and so on.
3. The idea of classified discussion: In mathematics, we often need to investigate under different circumstances according to the different nature of the research object. This classified thinking method is an important mathematical thinking method and an important problem-solving strategy.
4. undetermined coefficient method: When the mathematical formula we are studying has a certain form, we only need to find the value of the letter to be found in the formula to determine it. Therefore, substituting the known conditions into the formula with undetermined form will often produce an equation or equation group with undetermined letters, and then solving this equation or equation group can solve the problem.
5. Matching method: try to construct an algebraic expression into a plane, and then make the necessary changes. Matching method is an important deformation skill in junior high school algebra, which plays an important role in decomposing factors, solving equations and discussing quadratic functions.
6. Substitution method: in the process of solving problems, take one or several letters of the formula as a whole and use a new letter to represent it, thus further solving problems. Method of substitution can simplify a complicated formula and turn the problem into a more basic problem than the original one, so as to achieve the purpose of simplifying the complex and turning the difficult into the easy.
7. Analysis method: When researching or proving a proposition, the conclusion is traced back to the known conditions. From this conclusion, the sufficient conditions for its establishment are derived. If the establishment of this condition is not obvious, then take it as a conclusion and further study the sufficient conditions for its establishment until the known conditions are reached, so that the proposition can be proved. This kind of thinking process is usually called. Looking for reasons?
8. synthesis method: when studying or proving a proposition, if the direction of reasoning is to proceed from known conditions and draw conclusions step by step, this thinking process is usually called? Cause leads to result?
9, deductive method: from general to special reasoning method.
10, induction: reasoning method from general to special.
1 1. analogy: among many objective things, some things have similar properties. Between two or two things, according to their similarity or similarity in some attributes, the reasoning method that they may be the same or similar in other attributes is deduced. Analogy can be special to special, or general to general reasoning.
First of all, you should be very familiar with the contents involved in the exercises, explain the concepts clearly, and be very familiar with definitions, formulas, theorems and rules. You should know that solving and doing problems is only a part of the learning process, not the whole learning. You can't solve a problem for the sake of solving it. Solving problems is for reading. Is to check whether you have read the textbook, whether you have a deep understanding of the concepts, theorems, formulas and rules, and whether you can use these concepts, theorems, formulas and rules to solve practical problems. When solving problems, the clearer our concepts are, the more familiar we are with formulas, theorems and laws, and the faster we will solve problems. Therefore, before solving problems, we should familiarize ourselves with, remember and distinguish these basic contents by reading textbooks and doing simple exercises, correctly understand the essence of their meanings, and then do the following exercises all the time. I instruct students to learn in this way, and almost all students have greatly improved the speed of solving problems, with good results.
Second, we should be familiar with the knowledge we have learned before and the knowledge related to other disciplines involved in the exercises. For example, sometimes, we encounter an exercise that we can't do, not because we haven't learned what we want to learn now, but because we want to use a formula we learned in the past, but we can't remember it clearly; Or a physical concept to be used in math problems, we are not very clear; Or we need to use a special theorem, but we have never learned it, which greatly reduces the speed of solving problems. At this time, it is necessary to add some relevant knowledge that must be added first, and explain the concepts, formulas or theorems related to the topic clearly, and then solve the problem, otherwise it is a waste of time. Of course, the speed of solving problems is even more impossible.
Third, we should be familiar with the basic steps and methods of solving problems. The process of solving problems is a process of thinking. For some basic and common problems, predecessors have summarized some basic problem-solving ideas and common problem-solving procedures. Generally, as long as you follow these ideas and steps, it is often easy to find the answers to the exercises. Otherwise, detours will take more time.
Fourth, learn to sum up. After solving a certain number of exercises, the knowledge involved and the methods of solving problems are summarized, which makes the thinking of solving problems clearer and achieves the effect of giving inferences by analogy. Similar exercises can be seen at a glance, which can save a lot of time in solving problems.
Fifth, we should make practice easy first and then difficult, and gradually increase the difficulty of practice. The process of people's understanding of things is from simple to complex, from the outside to the inside step by step. A person's ability is also gradually increased through exercise. If more simple problems are solved, the concepts are clear and the formulas, theorems and solving steps are familiar, jumping thinking will be formed when solving problems, and the speed of solving problems will be greatly improved. When you get into the habit and encounter general problems, you can also maintain a high speed of solving problems. However, some of our students don't pay much attention to these basic and simple exercises and think it is unnecessary to spend time solving these simple exercises. As a result, the concept is unclear, formulas, theorems and problem-solving steps are unfamiliar, and there is nothing to be done when encountering a slightly more difficult problem, let alone the speed of solving problems.