First, primary school mathematics problem-solving methods: thinking in images.
Thinking in images means that people use thinking in images to understand and solve problems. Its thinking foundation is concrete image, and its thinking process is developed from concrete image.
The main means of thinking in images are objects, figures, tables and typical image materials. Its cognitive feature is that it is average in individual performance and always retains its intuition about things. Its thinking process is represented by representation, analogy, association and imagination. Its thinking quality is manifested in the active imagination of intuitive materials, the processing and refining of appearances, and then the essence, law or object are revealed. Its thinking goal is to solve practical problems and improve thinking ability in solving problems.
1, physical demonstration method
Demonstrate the conditions and problems of mathematical problems and the relationship between them with the physical objects around you, and analyze and think on this basis to find a solution to the problem.
This method can visualize the content of mathematics and concretize the quantitative relationship. For example: the problem of meeting in mathematics. Through physical demonstration, we can not only solve the terms of "simultaneity, relativity and encounter", but also point out the thinking direction for students. Another example is the problem of planting trees around a round (square) pond If you can do an actual operation, the effect will be much better
Chickens and rabbits are in the same cage. Make three tables: the first table is an example method one by one. According to the situation of 20 chickens and rabbits, assuming that there is only 1 chicken, there are 19 rabbits and 78 legs ..... so list them one by one until you find the desired answer; In the second table, after several enumerations, the rule of counting only and the number of legs is found, thus reducing the enumeration times; The third table is listed from the middle. Because there are 20 chickens and rabbits, each chicken is taken as 10, and then the marketing direction is determined according to the actual data.
Step 4 explore methods
According to a certain direction, trying to explore the law and explore the way to solve problems is called inquiry method. Hua, a famous mathematician in China, said that in mathematics, "the difficulty lies not in having a formula to prove it, but in how to find it before there is no formula." Suhomlinski said: In people's hearts, there is a deep-rooted need to become discoverers, researchers and explorers, and this need is particularly strong in children's spiritual world. "Learning to explore.
"People-oriented" is one of the basic concepts of the new curriculum. When it is difficult for people to turn a problem into a simple, basic, familiar and typical problem, a good way is often to explore and try.
First, the direction of inquiry should be accurate, the interest should be high, and random attempts or formalistic inquiries should be avoided. For example, when teaching "Scale", the teacher created a teaching situation of "students give questions to test the teacher", and the teacher said, "Shall we take the exam now?" Hearing this, the students were surprised. Just when the students were puzzled, the teacher said, "Do you want to change the past examination method and let you test the teacher?" The students are very interested after listening. The teacher said, "This is a map. You can measure the distance between the two places with a ruler at will. I don't care. "
Can I tell you the actual distance between the two places quickly? Do you believe it? "So the students went to the stage to measure the number, and the teacher answered the corresponding actual distance one by one. The students were even more surprised at this moment and said in unison, "Please tell us quickly. How did you work it out? " The teacher said, "Actually, a good friend is helping the teacher in the dark. Do you know who it is? Want to know? " So it leads to the "scale" of the content to be studied.
Second, directional speculation, repeated practice, in the constant analysis and adjustment to find the law.
Third, the combination of independent inquiry and cooperative inquiry. Independent, have the time and space to think freely; Cooperation can complement each other in knowledge, complement each other in methods, and occasionally collide with the spark of wisdom.
5. Observation
Through a large number of concrete examples, the method of summarizing and discovering the general laws of things is called observation. Pavlov said, "You should learn to observe first. You will never become a scientist unless you learn to observe. "
The contents of primary school mathematics "observation" generally include: ① the changing law and position characteristics of numbers; ② Relationship between conditions and conclusions; (3) the structural characteristics of the topic; (4) The characteristics of graphics and the relationship between size and position.
For example, look at a set of formulas: 25×4=4×25, 62×111× 62, 100× 6 = 6× 100 ...
Requirements for "observation":
First, the observation should be meticulous and accurate.
Second, scientific observation. Scientific observation is permeated with more rational factors, and it is a purposeful and planned observation of the research object. For example, when teaching the knowledge of cuboids, we should orderly observe: (1) faces-shape, quantity and the relationship between faces; (2) Edge-the formation and number of edges, and the relationship between edges (the opposite edges are equal; There are four opposite sides; The edges of a cuboid can be divided into three groups); (3) Vertex-the formation and number of vertices. An important function of understanding vertices is to introduce the concepts of rectangular body length, width and height.
6. Typical method
According to the topic, the method of associating the problem-solving laws of the solved typical problems to find out the problem-solving ideas is called the typical method. Typical is relative to universal. To solve mathematical problems, some need general methods and some need special (typical) methods. Such as normalization, multiple ratio and induction algorithm, travel, engineering, eliminating similarities and seeking differences, averaging and so on.
When using the typical method, we must pay attention to:
(1) Master the key and laws of typical materials.
(2) Be familiar with typical materials, and be able to quickly associate them with applicable models, so as to determine the required problem-solving methods.
(3) Typical is associated with skill.
7. Scaling method
The method to solve the problem by estimating the scale of the studied object is called scale method. The scaling method is flexible and ingenious, but it depends on the expanding ability and imagination of knowledge.
Idea 1: "Zoom in". Through observation, it is found that the scores of Chinese, mathematics and foreign languages appear twice in the topic. We require a total of197+199+196, which is "twice the foreign language score". Divide by 2 to get the sum of three subjects, and then subtract any two subjects to get the third subject.
Idea 2: "Shrink". We subtract the score outside the language from the sum of the scores, 199- 197=2 (points), which is the difference between the scores of mathematics and English. The sum of math and English is 196, so it is not difficult to get math scores again.
Scaling method is sometimes used in estimation and checking calculation.
8. Verification method
Is your result correct? You can't just wait for the teacher's judgment. It is important to have a clear mind and a clear evaluation of your own study, which is an essential learning quality for excellent students.
Verification method has a wide range of applications and is a basic skill that needs to be mastered skillfully. Through practical training and long-term experience accumulation, I constantly improve my verification ability and gradually develop a good habit of being rigorous and meticulous.
(1) is verified in different ways. Textbooks have repeatedly suggested that subtraction is tested by addition, subtraction, multiplication and division.
(2) Substitution test. Is the result of solving the equation correct? See if both sides of the equal sign are equal by substitution. You can also use the result as a condition for reverse calculation.
(3) Whether it is practical. Mr. Tao Xingzhi's words, "A thousand teachers teach people to seek truth, and ten thousand teachers learn to be human beings", should be implemented in teaching. For example, it takes 4 meters of cloth to make a suit, and the existing cloth is 3 1 meter. How many suits can you make? Some students do this: 3 1÷4≈8 (set)
It is undoubtedly correct to keep the approximate figures according to the rounding method, but it is not realistic, and the rest of the cloth for making clothes can only be discarded. In teaching, common sense should be valued. The approximate calculation of the number of clothes sets should use the "tail cutting method".
(4) The motivation of verification lies in guessing and questioning. Newton once said, "Without bold speculation, there will be no great discovery." "Guess" is also an important strategy to solve the problem. It can develop students' thinking and stimulate the desire of "I want to learn". In order to avoid guessing, we must learn to verify. Verify whether the guess result is correct and meets the requirements. If it does not meet the requirements, adjust the guess in time until the problem is solved.
Second, primary school mathematics problem-solving methods: abstract thinking methods
The thinking process of reflecting reality with concepts, judgments and reasoning is called abstract thinking, also called logical thinking.
Abstract thinking is divided into formal thinking and dialectical thinking. Objective reality has its relatively stable side, and it can adopt the form of thinking; Objective existence also has its constantly developing and changing side, and we can adopt dialectical thinking. Formal thinking is the basis of dialectical thinking.
Formal thinking ability: analysis, synthesis, comparison, abstraction, generalization, judgment and reasoning.
Dialectical thinking ability: contact development and change, law of unity of opposites, law of mutual change of quality, law of negation of negation.
Mathematics in primary and secondary schools should cultivate students' preliminary abstract thinking ability, focusing on:
(1) The thinking quality should be agile, flexible, connected and creative.
(2) In the way of thinking, we should learn to think methodically and systematically.
(3) in terms of thinking requirements, the thinking is clear, the cause and effect are clear, the words must be reasonable, and the reasoning is strict.
(4) In thinking training, we should require correct application of concepts, proper judgment and logical reasoning.
9. Inspection method
How to correctly understand and apply mathematical concepts? The common method of primary school mathematics is comparison. According to the meaning of mathematical problems, the method of solving problems through understanding, memorizing, identifying, reproducing and transferring mathematical knowledge is called contrast method.
The thinking significance of this method lies in training students to correctly understand, firmly remember and accurately identify mathematical knowledge.
10, formula method
Methods to solve problems by using laws, formulas, rules and rules. It embodies the deductive thinking from general to special. Formula method is simple and effective, and it is also a method that primary school students must learn and master when learning mathematics. But students must have a correct and profound understanding of formulas, laws, rules and regulations, and can use them accurately.
1 1, comparison method
By comparing the similarities and differences between mathematical conditions and problems, we study the reasons for the similarities and differences, so as to find a solution to the problem, which is the comparative method.
Comparative law should pay attention to:
(1) Finding similarities means finding differences, and finding differences means finding similarities, and being indispensable means being complete.
(2) Find the connection and difference, which is the essence of comparison.
(3) Comparison must be conducted under the same relationship (same standard), which is the basic condition of "comparison".
(4) To compare the main contents, try to use the "exhaustion method" as little as possible, which will make the key points less prominent.
(5) Because of the rigor of mathematics, comparison must be meticulous, and often a word or a symbol determines the right or wrong conclusion of comparison.