Difficulties in primary school mathematics teaching _ On the difficult teaching of primary school mathematics

First, the meaning of teaching difficulty What is teaching difficulty? Some scholars believe that teaching difficulty generally refers to the knowledge content that is difficult for teachers to explain clearly, difficult for students to understand or easy to make mistakes. Some scholars believe that the difficulty in mathematics refers to the knowledge that students are not easy to understand or the skills that students are not easy to master. According to the author's understanding, teaching difficulties can be determined from two aspects: basic knowledge and basic skills, that is, concepts, principles, laws and formulas that are difficult for students to understand can be considered as difficulties, and those that are difficult to solve some practical problems by using basic knowledge or internalize through repeated training can also be considered as difficulties.

It should be noted that the difficulty is not necessarily the key point, and the key point is not necessarily the difficulty, but some contents are both difficult and important. The difficulty depends on the actual level of students. The same problem is difficult in this class, but not necessarily in another class.

Second, the emergence of teaching difficulties

Modern cognitive development theory holds that the development of students' cognitive structure is a process in which students constantly use assimilation and adaptation to reconstruct the original cognitive structure in the process of understanding new knowledge.

From the perspective of cognitive development theory, in teaching, if the learning content can be incorporated into the existing cognitive structure through students' thinking, thus enriching and strengthening the existing thinking tendency and behavior mode, such learning content is easy for students to understand. If the learned content conflicts with students' existing cognitive structure and new information, which leads to the adjustment of the original cognitive structure, it is necessary to establish a new cognitive structure. This knowledge of building a new cognitive structure through adaptation is more difficult. Because the cognitive structure itself has a formula, the negative effect of this formula will hinder the leap of cognition, thus causing difficulties in learning new knowledge and forming difficulties in teaching. Therefore, the difficulty of teaching depends to some extent on the content of teaching materials as cognitive objects, but also on students as cognitive subjects and teachers who play a leading role in teaching, that is, on the quality and ability of teachers and students.

Of course, in the process of learning the same content, assimilation and adaptation are often carried out at the same time, and it is difficult to completely separate them. Due to the differences of students' individual mathematical cognitive structure, the formation of teaching difficulties is bound to be different. In practice, we should flexibly determine the teaching difficulties according to the actual level of students.

Third, the breakthrough of teaching difficulties

1? Methods of inspiration and explanation. It is necessary for teachers to make a meaningful "talk" about the knowledge that students are difficult to understand. It should be noted that the "talk" here is not "indoctrination", but "inspiration and explanation", so that students can understand knowledge in a short time. This is the method we often use.

For example, it is difficult for students to understand the relationship between the number of trees planted and the interval in the first volume of the fourth grade experimental textbook of the Jiangsu Education Edition. To this end, I use the method of inspiring and explaining to teach, and the effect is good.

Teacher: (Multimedia shows pictures of rabbits and mushrooms in the example) Let's look at this picture together. How are rabbits and mushrooms arranged in the picture?

Health: Arrange according to the rule that a rabbit is followed by a mushroom.

Teacher: You speak very well! This is a question of interval arrangement, first the rabbit, and finally the rabbit. Like this, the rabbit is at the beginning and at the end. We regard rabbits as "objects at both ends" and mushrooms as "objects in the middle".

Teacher: Who can tell me how many rabbits there are? How many mushrooms are there?

Health: 8 rabbits and 7 mushrooms.

Teacher: (showing the fence map) Let's look at the fence map here again. Look carefully. What are the objects at both ends of this map? What is the object in the middle?

Health: The objects at both ends are stakes, and the objects in the middle are fences.

Teacher: Count the stakes and fences.

Health: stake 13, fence 12.

Teacher: (showing pictures of handkerchiefs) Let's look at the objects at both ends and the objects in the middle in this picture.

Health: The objects at both ends are clips, and the objects in the middle are handkerchiefs.

Teacher: How many clips and handkerchiefs are there?

Health: There are 10 clips and 9 handkerchiefs.

Teacher: Please fill in the number of objects at both ends and in the middle of the three pictures you just observed in the table below.

The teacher shows the following table. The numbers in the table are for students to fill in. )

Teacher: Please observe the table carefully. Can you find any patterns from it?

Health: I found that there are more 1 objects at both ends than in the middle.

Teacher: On the other hand, what can I say?

Health: There are fewer objects in the middle than at both ends 1.

Under the guidance of the teacher's inspiration, students have found the rules and made breakthroughs in teaching difficulties.

2? Demonstrate the experimental method. That is, using demonstration experiments to break through teaching difficulties. Demonstration experiment can make students observe and think from the dynamic operation process, so as to achieve the purpose of understanding knowledge.

For example, "In a cylindrical bucket with a bottom radius of 30 cm, a section of cylindrical steel with a radius of 10 cm is completely immersed in water. When the steel is taken out of the water, the water level in the bucket drops by 5 cm. How long is this steel? " The teaching difficulty of this problem is to make students understand that the volume of steel is actually the volume of water falling. How to establish the relationship between "the volume of steel" and "the volume of water falling" is a difficult problem for students. To this end, I instruct students to observe the experiment in teaching: put a section of cylindrical steel into a cylindrical beaker filled with water, so that the cylindrical steel is completely immersed in the water, so that students can observe the demonstration process. The teacher took the steel out of the beaker and asked the students to observe the changing process of the water surface and think about the following questions: Where is the water surface when the steel is not taken out? What happened to the water surface after the steel was taken out? Why is there such a change? What is the relationship between the volume of steel and the volume of water falling?

Through observation and thinking, the students found that after the steel was taken out, the underwater part of the beaker was a small cylinder, and the volume of this small cylinder was equal to the volume of cylindrical steel. In this way, the students successfully solved the volume problem of cylindrical steel, and then quickly calculated the length of steel: 3? Ω 14×302×5÷(3? 14× 102), the problem is solved.

3? Use metaphors. Although students can remember some basic knowledge and use what they have learned to solve some simple problems, it is sometimes unclear for them to tell the truth, which shows that students have not really understood it. Therefore, I often use metaphors to help students understand knowledge in teaching.

For example, students have some difficulties in understanding the concepts of "solution of equation" and "solution of equation", and sometimes they are confused. In order to make students understand these two concepts, I first ask students to find the value of X in x+20= 100, 23x=69, x- 13=50, and substitute the obtained value into the original equation to test, so as to guide students to observe whether the left and right sides of each equation are equal, and abstract the concept of "the solution of the equation". At the same time, the explanation is just like just now. Finally, inspire students to say a complete concept. Then draw inferences from one another. Put a pebble (weight 10g) on one side of the balance. If you want to know its weight, you need to open the weight box and find a weight equal to that of pebbles on the other side of the balance, so that it can be balanced left and right. Then, the weight of 10 gram is the "solution of the equation", and the process of finding the weight out of the box is the "solution of the equation".

4? Conversion narrative method. That is, the method of changing narrative form is adopted to reduce the difficulty and break through the difficulty. We often say "thinking mode". Indeed, students sometimes have fixed thinking, and some "standard form" problems can be solved smoothly, but slightly changed materials are difficult. When encountering such a situation, if the teacher can change the narrative form in time and let the students feel it in comparison, they will be inspired and solve the problem.

For example, "a project, built by team A, takes 20 days to complete, and built by team B, takes 30 days to complete. The two teams worked together for a few days, and the remaining team A completed the whole project in five days. How many days have the two teams been working together? " It is difficult for students to understand the expression in the question, which interferes with the thinking of solving the problem. In order to break through the difficulties, the narrative form of this problem can be changed to: "A project, undertaken by a construction team, takes 20 days to complete, and undertaken by a construction team takes 30 days to complete. Now it's a project team for five days, and the rest are a team and b team. How many days have Team A and Team B been working together? "

Obviously, although these two questions are expressed differently, their essence is the same. Therefore, the problem was quickly solved:

Set the quantity calculation method. That is, using the method of setting numbers as an example, the problem is solved by calculation. Some problems seem to lack conditions and are difficult to solve. At this time, if we use the fixed number method, we can quickly find a solution to the problem.

For example, "A is 25% more than B, and B is a few percent less than A?" If the number of B is 100, then the number of A is100× (1+25%) =125, so we can quickly find out the percentage of B less than A: (125-1. 2=20%.

This article is aimed at users who have not installed a PDF browser. Please download and install the full text of the original text first. Of course, we can also directly use letters to indicate the numbers to be set for some questions.

For example, "in a math exam, the average score of a class is 78, and the average score of boys and girls is 75? 5 points and 8 1 minute. What is the ratio of boys to girls in this class? "

We can assume that the boy is X and the girl is Y, so 75? X+8 1y = 78 (x+y) is simplified to 3y=2? 5x, that is, x∶y=6∶5, that is, the ratio of boys to girls in this class is 6∶5.

6? Drawing observation method. It is a problem-solving strategy to let students break through difficulties by drawing lines.

For example, "Party A and Party B each walked from AB at a certain speed, and met for the first time at at 500, the starting point of Party A ... After meeting, everyone will move on, reach the starting point of the other party and then turn back. The second meeting will be 300 meters from the starting point of B. How many meters are the two places apart? "

Draw the following line segment diagram and you will soon find a solution to the problem. As can be seen from the figure, Party A and Party B walked the whole distance, and Party A walked 500 meters. In the whole process, Party A and Party B walked three full distances, that is, Party A walked (500×3) meters and walked 300 meters more, so the distance between the two places was 500× 3-300 =1200m.

7? Comparative analysis method. "Comparison is the basis of all understanding and thinking. It is through comparison that we understand everything in the world. " There are many connections and differences in primary school mathematics (ushinski). Making full use of comparative method in teaching is helpful to break through the teaching difficulties, prevent knowledge confusion and improve the ability of discrimination.

For example, find the perimeter (unit: cm) of the following figure (figure 1).

Many students feel that this problem lacks conditions and cannot be solved for a while. At this time, you can present a rectangle (Figure 2) for students to observe and think by comparing the two figures: What do you think is the circumference of the original figure by comparing these two figures? Then do a dynamic demonstration, move two horizontal segments upward to connect with the top horizontal segment, and then move two vertical segments to the right to connect with the rightmost vertical segment. At this point, the students suddenly realized that the circumference of this figure can be calculated as follows: (10+5)×2.

8? Clever use of transformation method. The so-called transformation is to transform the original problem into a problem that can be solved or easily solved as much as possible. It is characterized by changing the difficult to the easy, the general to the special, the special to the general, the compound to the single, and the recessive to the dominant. Therefore, timely and appropriate use of the conversion method can not only break through the difficulties, but also help students form correct and flexible ideas and improve their ability to analyze and solve problems.

For example, there is an old classic title: "Legend has it that there was a wealthy businessman in Arabia who left a will when he died: After my death, I gave 17 horse to three sons. The eldest son gets the total number of horses, the second son gets the total number of horses, and the third son gets the total number of horses, but it is not allowed to kill or sell horses. After the death of a wealthy businessman, three sons and relatives could not share horses. Now please help me divide these horses.

To solve this problem, if you don't think of the idea of "borrowing horses to get points", the result of getting points will not be an integer result. To this end, I made the following tips: Can the three scores in the question be converted into forms related to the ratio? Then organize students to explore cooperatively. With everyone's efforts, it is thought that if you borrow a horse, you can convert the three points in this problem into ratios, that is, the ratio of the number of horses shared by three sons is:: = 9: 6: 2, and then use the idea of proportional distribution to solve the problem: the eldest son gets 17×=9 (horse) and the second son gets17× =.

In mathematics teaching, there are many ways to break through the difficulties. As long as you are good at thinking and teaching according to students' cognitive characteristics, you will break through the difficulties in teaching.

Author unit

Jiangsu province Suzhou industrial park Xincheng garden primary school

Editor in Charge: Cao Wen

Note: "Please read the charts, formulas and notes in PDF format involved in this article."

This article is the full text of the original. Users who don't have a PDF browser should download and install the original text first.