What are the means of mathematics teaching?
First, pay attention to the learning environment and let students participate in mathematics teaching.
In the discussion class, teachers carefully design discussion questions, give reasonable guidance, and discuss and learn among students.
Second, pay attention to the problem situation and let students get close to mathematics.
In mathematics teaching, teachers should carefully create problem situations, stimulate students' enthusiasm for learning new knowledge, narrow the distance between students and new knowledge, and let students get close to mathematics.
Third, pay attention to hands-on operation and let students experience mathematics.
Teachers design mathematics teaching as tangible materialized activities, so that students can have a clear understanding of very abstract knowledge, and the experience gained by students through hands-on operation is very profound.
Fourth, pay attention to independent exploration and let students? Reengineering? mathematics
When students have doubts about what they are interested in and are eager to know the mystery, teachers should not simply teach what they know directly to students, but should fully believe in students' cognitive potential, encourage students to explore independently, and actively engage in mathematical activities such as observation, experiment, guess, reasoning and communication, so as to boldly? Reengineering? mathematics
Fifth, emphasize the application of life and let students practice mathematics.
In teaching, teachers should often let students use what they have learned to solve practical problems in life, so that students can master what they have learned in time in the process of practicing mathematics, such as explaining the stability of triangles, the instability of parallelograms and the rotation invariance of circles with mathematical knowledge.
A summary of mathematics teaching methods in primary schools
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 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. Not only can it be solved through physical demonstration? Walk in opposite directions at the same time, meet? And so on, and pointed 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
In the second grade math textbook, three children meet and shake hands. Every two people shake hands. * * * How many times do I have to shake hands? With what? Put three different digital cards into two digits. * * * How many seats can you put? . If such permutation and combination knowledge is demonstrated in kind, it is difficult to achieve the expected teaching goal in primary school teaching.
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 kept well before use. This can effectively improve classroom teaching efficiency and students' academic performance.
Performance.
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 human processing and arrangement of representations. Once the graphic method is inconsistent with the actual situation, it is easy to make the association and imagination on this basis appear fallacy or go into misunderstanding, which will eventually lead to wrong results. For example, some math teachers love to draw mathematical figures by hand, which will inevitably lead to inaccuracies and misunderstandings among students.
In classroom teaching, we should use graphic methods to solve problems. Some topics, pictures come out, and the results come 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 thinking, 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?
Example 2. Judgment: In the isosceles triangle, 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: compare the area first, and then compare the circumference.
Idea: Make an auxiliary line. Figure A occupies a large area and Figure B occupies a small area. So? The area of Figure A is larger than that of Figure B? Is correct. Straight advertising is shorter than curve advertising, 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 the law, and is 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 rules. For example, positive-negative ratio, sorting out data, multiplication formula, numerical order, etc. are most used in teaching? List method? .
Solving the traditional mathematical problem with list method: the problem of chickens and rabbits in the same cage. Make three tables: the first table is an example. According to the conditions of 20 chickens and rabbits, assuming that there are only 1 chicken, there are 19 rabbits and 78 legs, and 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, 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 should be centered on inquiry? , 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. Like teaching? Scale? When did the teacher create it? Students give questions to test teachers? Teaching situation, teacher:? Now, do we have a good exam? As soon as the students heard this, it was strange. Just when the students were puzzled, the teacher said, Today, I changed the way I used to take exams, and you tested the teacher with questions. Would you like to? The students are very interested after listening. The teacher said, here 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. Can you believe it? So the students went to the stage to measure the report, and the teacher answered the corresponding actual distance one by one. The students were even more surprised at this moment and said in unison, Teacher, 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 about it and then draw out what you want to learn? Scale? .
Second, directional speculation, repeated practice, in the constant analysis and adjustment to find the law.
Example 3. Find a rule to fill numbers.
( 1) 1、4、 、 10、 13、 、 19;
(2)2、8、 18、32、 、72、 。
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.
In primary school mathematics teaching activities, teachers should try their best to create inquiry scenes for students, create opportunities for students to explore and encourage students to have the spirit and habit of inquiry.
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. Unless you learn to observe, you will never become a scientist." ?
Elementary school math? Observation? The general contents are as follows: 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, observe a set of formulas: 25? 4=4? 25,62? 1 1= 1 1? 62, 100? 6=6? 100 summary multiplication exchange rate: In the multiplication formula, the positions of the two factors are interchanged and the product remains unchanged.
? Observation? Requirements:
First, the observation should be meticulous and accurate.
Example 4. Find out what is wrong with the following questions 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)? five
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, what should we do when teaching the knowledge of cuboids? Batman? Observation: (1) Surface shape, quantity and the relationship between surfaces; (2) The formation and number of sides, and the relationship between sides (the opposite sides are equal; There are four opposite sides; The edges of a cuboid can be divided into three groups); (3) The formation and number of vertices. An important function of understanding vertices is to introduce the concepts of length, width and height of rectangles.
Third, observation must be combined with thinking.
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.
Exodus 7. It is known that the father is 30 years older than his son, and the father is exactly seven times his son this year. How old are father and son this year? The key is that 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.
Example 8. Do you see it? There is a bus line in a city with a length of 16500m, with an average stop every 500m. How many stops does this line need? It should have something to do with it. How many minutes does it take to saw wood? Typical problem.
(3) Typical is associated with skill.
Example 9. There are 82 engineering teams in Party A and Party B. If 8 people are transferred from Team B to Team A, the numbers of the two teams are exactly equal. How many people are there in Team A and Team B? Tip: The total number of the two teams has not changed before and after the adjustment. Calculate the adjusted number of teams first, 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 and imagination of knowledge.
Example 16. Find the least common multiple of 12 and 9.
What is the general method to find the least common multiple of two numbers? Short division? Method, which is based on the prime factors of these two numbers to find their least common multiple. 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? ; And 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 let's expand the application and enlarge it according to the typical method 2? Large number? Find the least common multiple of 12 and 9.
If 12 is not a multiple of 9, multiply 2 to get 24, which is still not a multiple of 9, and multiply 3 to get 36, which is a multiple of 9. Then, the least common multiple of 12 and 9 is 36. The key point of this method is that if a large number is not a multiple of a decimal, double the large number, but it must start from 2 times. If you expand 6 times at once, the numbers are their common multiples, not the smallest.
Example 17. In the final exam, the sum of Xiaogang's Chinese score and English score 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 the scores of Chinese, mathematics and foreign languages appear twice in the topic. Find the sum of197+199+196. What's the total? Twice the foreign language score? Divide by 2 to get the sum of three families, and then subtract any two families to get the third family.
Idea 2:? Shrinkage? . 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.
Example 18. Check whether the following calculation results are correct?
( 1) 18.7? 6.9= 137.3; (2) 17485? 6.6=3609.
For (1), use the global estimation and enlarge it to 19? 7= 133, and the estimate is less than 133, so the result of this problem is wrong. For (2) using the highest order estimation, let 17 be 18, and 6.6 be 6 18? 6=3, obviously the highest number of answers will not be 3, so the result of this question is incorrect.
Example 19. Put the chicken and rabbit together, * * * has 48 heads, 1 14 feet. 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 2 times. Then, the chicken has the same number of feet as its head, and the rabbit has twice as many feet as the rabbit. Therefore, after the total number of feet is reduced by 2 times, the difference between the total number of feet of chickens and rabbits and the 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 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. ? 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 (sets)
Follow? Rounding method? It is undoubtedly correct to keep the approximate figure, but it is not realistic. The rest of the cloth for making clothes can only be discarded. In teaching, common sense should be valued. How is the quantity of clothes calculated? Tailing method? .
(4) The motivation of verification lies in guessing and questioning. Newton once said: Without bold speculation, there is no great discovery. Guess what? It is also an important strategy to solve the problem. Can develop and stimulate students' thinking? I want to learn? Desire. 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, 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 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.
To cultivate students' preliminary abstract thinking ability in primary school mathematics, the key points are: (1) thinking quality, which 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.
Example 20. The sum of two 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.
Do you want to compare it here? Eliminate? And then what? Even number? These two mathematical concepts. Only by fully understanding these two concepts can we make a correct judgment.
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.
Example 22. Count 59? 37+ 12? 59+59
59? 37+ 12? 59+59
=59? (37+ 12+ 1) uses the multiplication distribution rule.
=59? Using addition calculation rules
=(60- 1) ? Composition rules of application number 50
=60? 50- 1? Using the law of multiplicative distribution
=3000-50 Apply multiplication calculation rules.
=2950 Use subtraction algorithm.
1 1. Comparative 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) comparisons must be made under the same relationship (same standard). What is this? Compare? The basic conditions of.
(4) Compare the main contents and use them as little as possible? Exhaustive method? By comparison, that will make the focus 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 original intention of this question is correct? What is the difference between the highest digit of a number and the highest digit of the decimal part? , what else? Numbers and values? Difference, etc.
Example 24. The sixth grade students planted a number of trees. If everyone plants 5 trees, there are 75 trees left. If each person plants 7 trees, there will be a shortage of 15 seedlings. 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): Everyone is 7-5=2 (tree), so 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 grouped into larger classes, and the larger classes are subdivided into smaller classes according to differences.
Classification is to pay attention to the different levels between categories and subcategories to ensure that subcategories in categories are not duplicated, omitted or crossed.
Example 25. According to the number of divisors, natural numbers can be divided into several categories.
A: 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.