Good methods can let us give full play to our talents, while poor methods may hinder them. -[English] Bernard.

"Mathematics provides language, ideas and methods for other sciences" and "initially learns to use mathematical thinking mode to observe and analyze the real society and solve problems in daily life and other disciplines". (Primary Mathematics Curriculum Standard)

There are two kinds of mathematical thinking methods, image thinking method and abstract thinking method.

Primary school mathematics should cultivate students' thinking ability in images and lay a solid foundation for developing abstract thinking ability.

First, 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 develops from concrete image.

The main means of thinking in images are objects, figures, tables and typical image materials. Its cognitive characteristics are average in individual performance and always keep intuition about things. Its thinking process is represented by representation, analogy, association and imagination. Its thinking quality is manifested in actively imagining intuitive materials, processing and refining representations, and then prompting the essence, laws or finding objects. Its thinking goal is to solve practical problems and improve itself 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 encounter problem in mathematics can not only solve the terms of "simultaneity, relativity and encounter" through physical demonstration, but also point out the thinking direction for students. For another example, the problem of planting trees around a round (square) pond will be much better if it can be operated in practice.

In the second grade math textbook, "When three children meet and shake hands, every two people shake hands once, * * * must shake hands several times" and "How many digits can * * * make up with three different digital cards". In primary school teaching, it is difficult to achieve the expected teaching goal if the knowledge of this arrangement and combination is demonstrated in kind.

Especially some mathematical concepts, if there is no physical demonstration, primary school students can't really master them. Learning the area of rectangle, understanding the cuboid and the volume of cylinder all depend on physical demonstration as the basis of thinking.

Therefore, primary school math teachers should make as many math teaching (learning) tools as possible, and these teaching (learning) tools should be well preserved and reused after use, which can effectively improve classroom teaching efficiency and students' academic performance.

Achievement.

2, graphic method

With the help of intuitive graphics, we can determine the direction of thinking, find ideas and find solutions to problems.

Graphic method is intuitive and reliable, easy to analyze the relationship between numbers and shapes, not limited by logical deduction, flexible and open-minded. However, the graphic method depends on the reliability of people's handling and arrangement of representations. Once the graphic method does not conform to the actual situation, it is easy to produce fallacies or misunderstandings in association and imagination based on it, which will eventually lead to wrong results. For example, some math teachers love to draw mathematical figures by hand, which will inevitably bring inaccuracies and misunderstandings to students.

In classroom teaching, we should use graphic methods to solve problems. Some questions and pictures came out and the results came out; Some questions have good pictures, and students will understand the meaning of the questions; For some problems, drawing can help to analyze the meaning of the problem and inspire ideas, as an auxiliary means of other solutions.

Example 1 It takes 24 minutes to saw a piece of wood into three sections, and how many minutes does it take to saw it into six sections? (Figure omitted)

The thinking method is: graphic method.

The thinking direction is: watch it several times for a few minutes at a time.

The idea is: how many times does it take to see the third paragraph in a few minutes and how many times does it take to see the sixth paragraph in a few minutes?

In the isosceles triangle of Example 2, point D is the midpoint of the bottom BC, the area of Figure A is larger than that of Figure B, and the perimeter of Figure A is larger than that of Figure B (omitted).

Thinking method: graphic method.

Thinking direction: first compare the area, then the circumference.

Idea: Make an auxiliary line. The area in Figure A is large, but the area in Figure B is small, so "the area in Figure A is larger than that in Figure B" is correct. The line segment AD is shorter than the curve AD, so "the circumference of Figure A is longer than that of Figure B" is wrong.

3. List method

The method of analyzing, thinking, looking for ideas and solving problems through lists is called list method. List method is clear, easy to analyze and compare, prompt rules, and also beneficial to memory. Its limitation lies in the small scope of solution and narrow applicable problems, which are mostly related to finding or displaying the law. For example, the teaching of positive and negative proportions, sorting out data, multiplication formulas, numerical order, etc., mostly adopts the list method.

Solve the traditional mathematical problem: the problem of chickens and rabbits in the same cage. Make three tables: the first table is an example. According to the situation of 20 chickens and rabbits, if 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 laws and methods to solve problems is called inquiry method. Hua, a famous mathematician in China, said that in mathematics, "the difficulty lies not in the proof of formulas, but in how to find formulas before there are no formulas." Suhomlinski said: In people's hearts, there is a deep-rooted need to be a discoverer, researcher and explorer, but in children. This demand is particularly strong. "Learning should focus on inquiry" is one of the basic concepts of the new curriculum. When it is difficult for people to turn problems into simple, basic, familiar and typical problems, they often take a good way to explore and try.

First, the direction of inquiry should be accurate and the interest should be high. Arbitrary attempts or formalistic inquiries are prohibited. For example, when teaching "Scale", the teacher created a teaching situation of "students give questions to test their teachers", and the teacher said, "Do we have to test 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. The teacher said, "This is a map. You can measure the distance between the two places with a ruler at will, and I can tell you the actual distance between the two places quickly. Do you believe it? " So the students took the stage to measure and count, and the teachers answered the corresponding actual distances one by one. At this time, the students were even more surprised and said in unison, "Teacher, please tell us how you worked 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? " Then introduce the scale of the content to be studied.

Second, directional speculation, repeated practice, in the constant analysis and adjustment to find the law.

Example 3 Find a rule to fill in the numbers.

( 1) 1、4、 、 10、 13、 、 19;

(2)2、8、 18、32、 、72、 .

Third, the combination of independent inquiry and cooperative inquiry. Independent, free to think about time and space; Cooperation can complement each other in knowledge, complement each other in methods, and occasionally collide with the spark of wisdom.

In primary school mathematics teaching activities, teachers should try their best to create situations and opportunities for students to explore and encourage students to have the spirit and habit of exploring.

5. Observation

Through a large number of concrete examples, the method of discovering the general law of things is called observation. Pavlov said: "You should learn to observe first. If you don't learn to observe, you will never become a scientist."

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, size and position of graphics.

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.

Example 4 Find out where the following questions are wrong and correct them.

( 1)25× 16=25×(4×4)=(25×4)×(25×4);

(2) 18×36+ 18×64=( 18+ 18)×(36+64)

Example 5 Write the numbers of the following questions directly:

( 1)3.6+6.4 (2)3.6+6.04

(3) 125×57×0.04 (4)(35 1-37- 13)÷5

Second, scientific observation. Scientific observation permeates more rational factors and treats the research object purposefully and in a planned way. For example, when teaching the knowledge of cuboids, we should observe the "order": (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.

Third, observation must be combined with thinking.

Example 6

seven

10

six

18

This is a thinking question for the next semester. If you just observe and don't think, you don't know what to do with this problem.

6. Typical method

The method of associating the problem-solving rules of the solved typical problems according to the topics, so as to find out the problem-solving ideas is called the typical method. Typical is relative to universality. To solve mathematical problems, some need general methods, and some need special (typical) methods, such as normalization, multiplication and induction, travel, engineering, eliminating similarities and differences, averaging and so on.

When using the typical method, we must pay attention to:

(1) Master the key and laws of typical materials.

It is known that the father is 30 years older than his son, and the father is just seven times older than his son this year. How old are the father and son this year? The point is: the father is 30 years older than his son, and the father is several times older than his son. Typical problems have typical solutions. To really learn mathematics well, we must understand and master general ideas and solutions, and learn typical solutions.

(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.

For example 8, see "There is a bus line in a city with a length of16500m, with an average stop every 500m. How many stops does this line need? " This topic should be related to the typical question mentioned above, "How many minutes does it take to saw wood?"

(3) Typicality is associated with skill.

Example 9 There are 82 engineering teams A and B. If 8 people are transferred from Team B to Team A, the numbers of the two teams are exactly the same. How many people are there in each team? Tip: The total number of teams has not changed before and after the adjustment. Calculate the adjusted number of teams and then calculate the original number of teams.

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 of knowledge and its imagination.

Example 16 Find the least common multiple of 12 and 9.

The general method of finding the least common multiple of two numbers is "short division", which is based on the prime factors of these two numbers. But there are also two typical methods: one is "if two numbers are prime numbers, then the least common multiple of these two numbers is their product"; Second, "If a large number is a multiple of a decimal, then the least common multiple of these two numbers is a large number". Now we use "large numbers" according to the typical method 2 to find the least common multiple of 12 and 9.

If 12 is not a multiple of 9, multiply 2 to get 24, but it is still not a multiple of 9, multiply 3 to get 36, and 36 is a multiple of 9. Then, the least common multiple of 12 and 9 is 36. The key point of this method is to double a large number if it is not a multiple of a decimal, but it must start from 2 times. If it unfolds immediately,

17 final exam, the sum of Xiaogang's Chinese and English scores is197; Chinese and math scores add up 199; The math and English scores add up to 196. Think about it, which subject Xiao Gang scored the highest? Can you work out the scores of Xiaogang's subjects?

Idea 1: "Zoom in". Through observation, it is found that there are two scores of three subjects in the topic, and the sum197+199+196 is "twice the score of other subjects". Divide by 2 to get the sum of three subjects, and then subtract any two subjects.

Idea 2: "Narrow down the scope". We subtract the sum of extra-lingual scores from math scores, 199- 197 = 2 (points), which is the difference between math and English scores. 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.

Example 18 Check whether the following calculation results are correct?

( 1) 18.7×6.9= 137.3; (2) 17485÷6.6=3609.

For (1), the overall estimate is enlarged to 19×7= 133, and the estimate is less than 133, so the result of this question is wrong. For (2), the highest estimate is used, and 17 is regarded as18,6.6.

Example 19 put the chicken and rabbit together, * * * has 48 heads, 1 14 feet, and ask how many chickens and rabbits there are.

This is a typical problem of chickens and rabbits in the same cage. We also use scaling method to reduce the number of feet of chickens and rabbits by two times, so that the number of feet of chickens is the same as that of rabbits, and the number of feet of rabbits is twice that of rabbits. Therefore, after the total number of feet is reduced by two times, the difference between the total number of feet of chickens and rabbits and their total number of feet is the number of rabbits.

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 repeatedly put forward that subtraction is tested by addition, subtraction, addition, multiplication, division and multiplication.

(2) Substitution test. Is the result of solving the equation correct? Use method of substitution to see whether the two sides of the equal sign are equal. You can also use the result as a condition for reverse calculation.

(3) Whether it is realistic. "Thousands of teachers in Qian Qian teach people to seek truth, and thousands of students in Qian Qian learn to be human." Mr. Tao Xingzhi's words 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. Attention should be paid to common sense in teaching, and the approximate calculation of the number of clothes should be made by the method of "removing the tail".

(4) The motivation of verification lies in guessing and questioning. Newton once said, "Without bold speculation, there is no great discovery." "Guess" is also an important strategy to solve problems, which can broaden students' thinking and stimulate students' desire for learning. In order to avoid blind guessing, we must learn to verify whether the guessing result is correct and meets the requirements. If it does not meet the requirements, it is necessary to adjust the guess in time until it is solved.

Second, the abstract thinking method

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 formal thinking can be adopted. Objective existence also has its 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.

Primary school mathematics should cultivate students' initial abstract thinking ability. The key points are: (1) thinking quality, which should be agile, flexible, connected and creative; (2) The way of thinking should be orderly and well-founded; (3) Thinking requirements, clear thinking, clear cause and effect, well-founded and strict reasoning; (4)

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 by comparing the meaning and essence of concepts, properties, laws, rules, formulas, nouns and terms and relying on the understanding, memory, identification, reproduction and transfer of mathematical knowledge is called comparison method.

The thinking significance of this method lies in training students to correctly understand, firmly remember and accurately identify mathematical knowledge.

Example 20. The sum of three consecutive natural numbers is 18, so what are the three natural numbers from small to large?

By comparing the concept of natural numbers with the properties of continuous natural numbers, we can know that the average sum of three continuous natural numbers is the middle number of these three continuous natural numbers.

Example 2 1, judgment: the number divisible by 2 must be even.

Here we should compare the two mathematical concepts of division and even number. Only by fully understanding these two concepts can we make a correct judgment.

10, formula method

The method of solving problems by using laws, formulas, rules and rules 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.

Example 22: Calculate 59×37+ 12×59+59.

59×37+ 12×59+59

= 59× (37+ 12+ 1) ............................................................................................................................................

= 59× 50 .........................................................................................................................................................................

= (60- 1) × 50 ........................................................................................................................................................

= 60× 50- 1× 50 .........................................................................................................................................................

= 3000-50 ......................................................................................................................................................................

= 2,950 ..........................................................................................................................................................................

1 1, comparison method

By comparing the similarities and differences between mathematical conditions and problems, we study the causes of similarities and differences, and then find a solution to the problem, which is the comparative method.

Comparative law should pay attention to:

(1) If you want to find similarity, you need to find difference. If you want to find difference, you need to find similarity. If you are indispensable, you need to be 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) We should grasp the main content and try to use the "exhaustive 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.

Example 23. Fill in the blanks: the highest digit of 0.75 is (), and the highest digit of the decimal part of this number is (); Compared with the fourth digit after the decimal point, their ()

Same, () different, the former is smaller than the latter ().

The purpose of this question is to distinguish between the highest digit of a number and the highest digit of the decimal part, as well as numbers and values.

Example 23, the sixth grade students planted a batch of trees, if each person planted five trees, the remaining 75 trees were not planted; If there are 7 trees per race, there will be a shortage of 15 saplings. How many students are there in the sixth grade?

This is a comparison between the two schemes. The similarities are: the number of sixth grade students remains unchanged; The difference is that the conditions in the two schemes are different.

Find a connection: the number of trees planted by each person has changed, and the total number of trees planted has also changed.

Solution (method): each person is 7-5=2 (tree), then the whole class is 75+ 15=90 (tree), and the class size is 90÷2=45 (person).

12, classification

As the saying goes, birds of a feather flock together.

According to the similarities and differences of things, things are divided into different categories, which is called classification. Classification is based on comparison. According to the similarity between things, they are combined into larger categories, and according to the differences, the larger categories are divided into smaller categories.

Classification should pay attention to the different levels between categories and subcategories to ensure that subcategories within categories are not duplicated, omitted or crossed.

Example 24. According to the number of divisors, natural numbers can be divided into several categories.

Answer: It can be divided into three categories. (1) A number with only one divisor is a unit number with only one number 1; (2) There are two divisors, also called prime numbers, and there are countless; (3) There are three divisors, also called composite numbers, and there are countless 1.

13, analysis method

A way of thinking that decomposes the whole into parts, decomposes complex things into parts or elements, and studies and deduces these parts or elements is called analytical method.

Foundation: The whole is made up of parts.

Thinking: In order to better study and solve the whole, we should first separate all the parts or elements of the whole, and then compare the requirements separately, so as to straighten out the problem-solving ideas.

That is to say, starting from the problem to be solved, we should correctly select two necessary conditions and deduce them in turn until the problem is solved. This problem-solving model is "tracing the cause from the result". Analysis is also called inverse deduction. Branch diagrams are often used to illustrate this idea.

Example 25: The toy factory plans to produce 200 toys every day, which has been produced for 6 days, and * * * has produced 1260 toys. How many toys exceed the standard on average every day?

Thinking: How many pieces do you need to exceed the plan every day? You should know: how many pieces are planned to be produced every day and how many pieces are actually produced every day. How many pieces are planned to be produced every day is known, and how many pieces are actually produced every day is not mentioned in the question. We must know how many days we actually produce and how many pieces we actually produce.

Branch diagram: (omitted)

14, comprehensive method

A method of combining all parts or aspects or elements of an object into an organic whole to study, deduce and think is called synthesis method.

When solving mathematical problems by comprehensive method, each problem is usually regarded as a part (or element). After analyzing the internal relationship between each part (or element) layer by layer, the problem requirements are gradually deduced. Therefore, the problem-solving mode of comprehensive method is dependent on cause, also known as forward deduction method. This method is suitable for mathematical problems with few known conditions and simple quantitative relationship.

Example 26. The difference between two prime numbers is a composite number less than 30, and their sum is a multiple of 1 1 and an even number less than 50. Write each set of numbers suitable for the above conditions.

Idea: Even numbers with multiples of 1 1 less than 50 at the same time are 22 and 44.

Both numbers are prime numbers, and the sum is even. Obviously, there is no 2 in these two prime numbers.

Two prime numbers whose sum is 22 are: 3 and19,5 and 17. Is their difference a composite of less than 30?

The two prime numbers with the sum of 44 are: 3 and 4 1, 7 and 37, 13 and 3 1 respectively. Is their difference a composite of less than 30?

This is the idea of comprehensive method.

15, equation method

Unknown numbers are represented by letters, and expressions (equations) containing letters are listed according to equivalence relations. The column equation is an abstract generalization process, and the solution of the equation is a deductive deduction process. The biggest feature of equation method is to treat the unknown as a known number and participate in the formulation and operation of formulas, which overcomes the deficiency that arithmetic method must avoid seeking knowledge formulas. It is beneficial to the transformation from known to unknown, thus improving the efficiency and accuracy of solving problems.

Example 27: A number is increased by 3 times 100, and then reduced by 2 times/36 to get 50. Find this number.

A barrel of oil used 40% for the first time and 10 kg for the second time, leaving 6 kg. How much does this barrel of oil weigh?

It is easier to solve these two problems with equations.

16, parameter method

A method of expressing related quantities by letters or numbers that only participate in formulas and operations without solving them, and listing formulas according to the meaning of the questions is called parameter method. Parameters are also called auxiliary unknowns and intermediate variables. Parametric method is the product of the extension and expansion of equation method.

Example 29. When a car climbs a mountain, the average speed is15km when it climbs the mountain and10km when it descends. What is the average speed of cars?

The average speed of going up and down the mountain can not be divided by the sum of the speeds of going up and down the mountain, but by the distance of going up and down the mountain.

For a job, A works alone for 4 days and B works alone for 5 days. How many days does it take for two people to do it together?

In fact, the total workload is regarded as "1", and this "1" is the parameter. If the total workload is regarded as "2, 3, 4 ...", this is only the most convenient operation.

17, exclusion method

The result of eliminating opposition is called exclusion.

The logical principle of exclusion is that everything has its opposite. In all kinds of right and wrong results, excluding all wrong results, the rest can only be correct results. This method is also called exclusion, screening or disproof. It is an indispensable formal thinking method.

Example 3 1. Why are all prime numbers odd except 2?

This requires the reduction to absurdity: all natural numbers greater than 2 are either prime numbers or composite numbers. Suppose a prime number greater than 2 has an even number, then this even number must be divisible by 2, that is, it must have a divisor 2. In addition to 1 and itself, there are other divisors (divisor 2), which must be a composite number rather than a prime number. This is different from the original assumption.

Judgment: (1) If two straight lines on the same plane are not parallel, they will intersect. (error)

(2) The numerator and denominator of a fraction are multiplied or divided by the same number, and the size of the fraction remains unchanged. (error)

18, special case method

For topics involving general conclusions, the method of solving problems by taking special values, drawing special pictures or setting special positions is called special case method. The logical principle of special case method is that the generality of things exists in particularity.

Example 33: The radius of a big circle is twice that of a small circle, the circumference is () times that of a small circle, and the area is () times that of a small circle.

You can take the radius of the small circle as 1, so the radius of the big circle is 2. After calculation, the correct result can be obtained.

Is the area of a square proportional to the length of its sides?

If the side length of a square is A and the area is S, then s:a=a = A (the ratio is uncertain).

Therefore, the area of a square is not proportional to its side length.

19, conversion method

Through a certain transformation process, the method to solve the problem by simplifying it into a typical problem is called simplification. Reduction is an important way of knowledge transfer and the first step to expand and deepen cognition. The logical principle of reduction is that things are generally related. Reduction is a common dialectical thinking method.

A pharmaceutical factory produced a batch of anti-SARS drugs, which was originally planned to be completed in 25 people 14 days. Due to urgent need, it had to be finished four days in advance. How many more people do you need?

This requires that when considering the problem, the "total working days" should be classified as "total workload"

The supermarket sent three kinds of vegetables: potatoes, tomatoes and cowpeas. Potato accounts for 25%, and the weight ratio of tomato to cowpea is 4: 5. Cowpea is known to be 36 kilograms more than potato. How many kilograms of tomatoes did the supermarket send?

It is necessary to classify "the weight ratio of tomatoes and cowpeas is 4: 5" as "what percentage of the total weight each", that is, to classify the application of proportion as the application of scores.