The new curriculum advocates that students' learning methods should be based on active participation, exploration and discovery, and exchanges and cooperation. The new curriculum reform attaches great importance to the cultivation of students' inquiry ability and inquiry learning; It is believed that the process of students' learning mathematics should be a rich and vivid thinking process in which students personally participate. Let students go through a process of practice and innovation. It can be seen that the new curriculum reform takes guiding students to carry out inquiry learning as one of the key points of the reform. Therefore, constantly create opportunities to guide students to learn to explore in cooperation and exchange. However, in actual teaching, there are some blindness and confusion in mathematical inquiry and cooperation activities. A considerable number of teachers have revealed that their interpretation of new ideas has deviated and even gone astray. Let me talk about my views on this issue.

First, the problems and countermeasures of inquiry learning in primary school mathematics classroom teaching

1. In inquiry learning, students' cognitive rules are not followed, and they are not tested from the perspective of students. Inquiry learning has become a behavioral process imposed on students.

2. Overemphasizing students' autonomy and weakening teachers' guiding role.

Under the new curriculum concept of independent inquiry as the main learning method, teaching activities should fully reflect students' autonomy in learning. However, due to some teachers' misunderstanding of students' subjective status, the classroom where the subject returns has been turned into a classroom where the subject is allowed to drift, which weakens the guiding role of teachers. It is mainly manifested in that students are allowed to explore independently without supervision, and some teachers dare not give guidance when organizing students to carry out mathematical inquiry activities, for fear of being labeled as "leading students by the nose"; Even with the guidance of teachers, it is impossible to grasp the timing and degree of intervention.

3. Pay attention to the behavior process of inquiry learning and ignore the development of thinking.

In mathematics teaching, we sometimes see the phenomenon that teachers provide students with some structured materials, then ask questions, describe the steps of inquiry, and finally let students use these materials to explore. After a class, the children did it and the conclusion was drawn smoothly. But is this kind of hands-on operation a mathematical inquiry? Some students are very happy from beginning to end, and their enthusiasm for participation is very high, but what students get is only superficial, so math class is not just to make students happy.

Through thinking, I think teachers should pay attention to the following points when designing the teaching content of inquiry learning:

1. We should look at mathematics from children's point of view, follow students' cognitive rules, and create a harmonious, democratic, vivid and lively atmosphere of inquiry learning for students. Constantly arouse students' cognitive conflicts, so that students can understand and master mathematics knowledge in the process of constantly overcoming thinking obstacles.

I remember that in March, 2005, I received a task from Mr. Tang Ruixiang, the former director of the teaching and research section, to give me an open class in the whole region. The theme of the activity is "old textbooks, reflecting new ideas" and the topic is "the volume of cones". When I received the task, I was very happy and at the same time felt a lot of pressure. Happily, I have another chance to challenge myself; Under pressure, how can we teach this open class well? It should not only be innovative, but also embody the new curriculum concept. Before that, I listened to several open classes of Volume of Cone. The teaching design is very routine, that is, first know the cone, and then show a group of models of cylinders and cones with equal bottom and equal height, so that students can observe that they are a group of cylinders and cones with equal bottom and equal height. Finally, by doing experiments, the sand was poured from the cone into the cylinder, and it was just filled three times, so it was concluded that the volume of the cone was one-third of that of a cylinder with equal bottom and equal height, and the teaching design of this idea was similar. When I was studying the textbook, I found such a problem. Generally, teachers in this class can draw conclusions through experiments, but according to students' cognitive rules, why do they suddenly compare the sizes of cylinders and cones? Why do you want to do this experiment? Unknown doing this experiment is a mathematical activity imposed on students by teachers, which does not follow students' cognitive laws at all, but is led by teachers. When thinking about this problem, I remembered an article I once read about the teaching of the volume of a cone. The teaching link designed by the author of this paper has solved this problem in my opinion. This is an article I happened to read in the magazine "Math Teachers in Primary and Secondary Schools" before. It's amazing to have such a clever teacher. Although I don't remember his name, I really admire that teacher's unique creativity. Thought of here, my heart sank again. If I follow the link designed by that teacher to attend this open class, I feel as if I have copied it. But I also think that applying the achievements of excellent teachers to my teaching is just a sign of learning from that teacher. So in the end, I borrowed the link designed by that teacher and achieved very good results in teaching. The teaching of cone volume is designed as follows: students learn new knowledge by guessing. First of all, through the courseware, let the students recall that a cylinder and a cone are three-dimensional figures formed by the rotation of a rectangle and a right triangle respectively. Displays a rectangle and a triangle. The long side of a rectangle is equal to the height of a right triangle, and the short side of a rectangle is equal to the base of a right triangle. The teacher asked: What is the relationship between the area of a rectangle and the area of a right triangle? Students answer that the area of a right triangle is half that of a rectangle. Then rotate the long side of the rectangle and the height of the right triangle respectively to get a cylinder and a cone. Please observe the cylinder and the cone and compare them. What is the relationship between them?

After observation, the students said that the base is equal and the height is equal. The teacher then asked: Please guess, what is the relationship between a cylinder with equal base and equal height and the volume of a cone? The picture is as follows:

Due to the influence of the previous comparison area, many students thought it was 1/2, some students guessed it was 1/3 through the imagination of space, and some students guessed it was 1/4. Is it 1/2, 1/3 or 65438? Then let the students do experiments with the materials given in their hands to verify their guesses. The design of this link really makes students realize the necessity of doing experimental verification. In this teaching process, suspense is set through interest to reveal contradictions and arouse students' cognitive conflicts, so that students will have doubts and curiosity. The ancients said, "Learning begins with thinking, and thinking originates from doubt." As long as the learning process of mathematics can slightly reflect the process of mathematical invention, we should give a reasonable guess a proper position. In teaching, let students make bold guesses and assumptions, put forward some premonitory ideas and realize instant epiphany, which is conducive to the development of students' creative thinking.

The design of the lesson "the volume of a cone" I just mentioned is to fully respect students' cognitive laws and let them go through the process of guessing, verifying and summarizing. Through this case, I thought a lot, especially about the problems that need to be paid attention to in my study. In inquiry learning, don't let students explore blindly, so it looks lively. In fact, students don't understand what's going on and willingly follow the teacher's ideas. This reminds me of Zhao Benshan's sketch Selling Kidnappers in the Spring Festival Evening. The teacher is selling kidnappers and the students are playing the role of buying kidnappers.

2. Teachers' reasonable "guidance" and students' clear exploration direction.

A good cognitive structure is the premise of students' exploration, and students' exploration is the expansion of life experience and existing knowledge. In teaching, teachers should help students communicate with each other constantly and establish knowledge networks. In teaching, teachers should consciously infiltrate some mathematical thinking methods, so that students can feel and understand flexibly, and guide students to constantly sum up thinking methods, thus enriching students' thinking experience and making students clear the direction of exploration. Therefore, in the process of students' inquiry, reasonable "guidance" is the key to the success of students' inquiry.

Not long ago, in the seminar class on campus, Huang Hui, a teacher from our school, gave a class on "Sum of the Interior Angles of Triangle". I think it is very successful to explore the teaching design of triangle interior angle sum, which well reflects the key role of teachers' "guidance". First of all, the teacher's courseware shows acute triangle, obtuse triangle and right triangle, and creates a problem situation in which they argue whose inner angle is bigger, and then lets students guess whose inner angle is bigger. Whose guess is right? First, use the actual measurement to verify it. Due to the error of measuring tool or measuring process, the sum of the internal angles of triangle is about 180 degrees. The teacher asked again: besides measuring, do you have any other way to verify the sum of the angles inside the triangle? The students thought of the method of tearing and folding. The teacher affirmed the students' ideas. Then the teacher asked: Students, what do you associate with 180 degrees? The classmate replied: Seeing 180 degrees, we thought 180 degrees was a straight line. It is the teacher's question that makes students connect new knowledge with old knowledge, so that in the process of tearing and folding, they naturally think of how to tear or fold the three angles of the triangle into a straight line, thus verifying that the sum of the internal angles of the triangle is 180 degrees. Without the teacher's questions, I think students have no direction to explore at all, just thinking hard there. Even if they want to tear or fold, they still can't consciously tear or fold into a straight line to verify whether the sum of the internal angles of the triangle is 180 degrees.

I still remember once listening to three math classes in the Wuhan Quality Class Competition, all about the circumference of a circle. In all three classes, students are asked to guess first. What does the circumference of a circle have to do with it? Through intuition, students guess that the circumference of a circle is related to its diameter. So, what is the relationship between the circumference and diameter of a circle? In two classes, students can measure the circumference and diameter of an object, and then calculate the ratio of circumference to diameter, so that the circumference is always more than three times the diameter. However, from the perspective of logical thinking, I want to ask: Why do we have to calculate the ratio of the circumference to the diameter of a circle? Why do we have to study the multiple relationship between the circumference and diameter of a circle? As a teacher, I have such doubts. Then we students must have such questions, but we just don't have the courage to ask them. In another class, the teacher's reasonable "guidance" just solved such a problem. The teacher also asked the students to guess what the circumference has to do with it. Through intuition, students guess that the circumference of a circle is related to its diameter. The teacher asked the students to fill in the form by measuring the circumference and diameter of the object. The teacher asked: What is the relationship between the circumference and diameter of a circle? The teacher explained that when studying the relationship between the two, it is generally based on "the relationship between sum, difference, product and multiple of division". Then ask the students to calculate the addition, subtraction, multiplication and division of the circumference and diameter of a circle through a calculator. There is no certain pattern. When the circumference of a circle is divided by the diameter, a certain rule is obtained: the circumference of a circle is always more than three times the diameter. It is concluded that there is a multiple relationship between the circumference and diameter of a circle. Through the above two cases, we can see that in inquiry learning, we should ask more why, look at mathematics with students' eyes, and teachers should reasonably "guide" and carefully design teaching links to improve the effectiveness of inquiry learning.

3. Provide students with an open learning space and sufficient exploration time.

Inquiry learning requires students to acquire the development of knowledge, skills, emotions and attitudes through inquiry activities. Therefore, it is necessary to provide students with suitable and open inquiry learning materials in teaching, so that students can enter a learning activity platform of free choice and independent discovery. Secondly, guide students to explore independently. In the process of exploring mathematical problems, students need to carefully observe, compare, guess, summarize, analyze and sort out repeatedly. This process can't be smooth sailing, but it will take some time to ensure. Sometimes, we see that in many inquiry learning classes, teachers ask students to cooperate in groups. Although they give students the opportunity to cooperate in inquiry, they often end up hastily because they have to complete the teaching task, which will inevitably reduce the learning effect of inquiry learning. Therefore, in order to improve the effectiveness of inquiry learning, teachers must provide students with sufficient inquiry time, so that students can truly understand the whole process of knowledge development and experience the joy of inquiry learning.

Create a situation similar to scientific research in teaching. Students can acquire knowledge and skills, develop their emotions and attitudes, especially explore the development of learning methods and learning processes of spirit and innovation through independent exploration activities such as finding problems, experiments, operations, investigations, information collection and processing, expression and communication. Therefore, if we want to carry out inquiry activities in teaching, teachers should make reasonable use and rational choice according to the teaching objectives and the actual situation of students. Only by correctly understanding inquiry teaching can inquiry learning play a better role in cultivating students' practical ability and innovative spirit.

Above, I briefly talked about my own ideas in teaching. We should look at the world with children's eyes, look at mathematics with children's eyes, follow students' cognitive rules, and create a harmonious, democratic and lively learning atmosphere for students. Clever creation of problem situations constantly causes students' cognitive conflicts, so that students can understand and master mathematics knowledge in the process of constantly overcoming thinking obstacles. And provide students with sufficient study space and time, so that students can spread their imagination wings, discover and explore in their study, and make mathematics classroom truly become a paradise for students to learn.

Second, the problems and countermeasures of inquiry and cooperative learning in primary school mathematics classroom teaching

Cooperative learning is a way of learning in which students can carry out learning activities together by using groups in teaching, so as to maximize the learning effect of themselves and others. Group cooperative learning has many advantages. Group cooperative learning can promote the construction of knowledge through the interaction and complementarity between students. Through cooperative learning, students' learning tasks change from individualization to the combination of individualization and cooperation, and the relationship between students changes from competition to cooperation and competition. Cultivating students' cooperative consciousness and ability can reduce their pressure, enhance their self-confidence and increase the opportunities for hands-on practice. So as to cultivate students' innovative spirit and practical ability and promote the personality development of all students.

However, there are many problems in group cooperative learning at present, such as: what kind of problems are suitable for group cooperative learning, what should be the relationship between group cooperative learning and independent thinking, how to deal with "centralized teaching" in group cooperative learning, and how teachers can play a regulatory role in cooperative learning. Let me talk about these problems and countermeasures.

1, randomness of cooperation process. The essence of group cooperation is that the appearance is close to the spirit, and students express their opinions, but they can't hear their peers' voices. So it is difficult to reach a consensus. Finally, the representatives in the group can only represent the ideas of some students. Such cooperation is very lively in form, but it is difficult to agree with it in actual effect. From this point of view, cooperative learning should have cooperative plans and steps, a clear division of labor, a certain degree of organization and discipline, and cooperative learning should also have certain cooperative rules.

2. The problem of choice is random, ignoring the problem of choosing suitable cooperative learning.

What problems should students cooperate to exchange? This is a realistic problem. Only when individuals encounter problems that cannot be solved independently or there are differences in ways and means to solve problems, will they have a strong desire to cooperate with others and listen to their opinions carefully. Unfortunately, however, there are very few problems that are really meaningful and need to be discussed in our math class. Most of the problems discussed are set by teachers to meet the needs of cooperation. Just let the students have something to talk about. For example, when a teacher asks a question, many students in the class raise their hands. However, the teachers completely ignored a pair of small hands held high and insisted on students' cooperation and communication. Besides wasting time, what effect can such cooperation and exchange receive? Since the implementation of the curriculum, many teachers have used "new" and "old" methods to evaluate whether students' autonomous learning, cooperative learning and inquiry learning run through the whole classroom. As a result, some teachers are misled by this idea and insist that students cooperate on issues that do not require cooperative learning. Therefore, teachers must grasp the "degree" of group cooperative learning in teaching, and they can neither learn everything cooperatively nor simply teach problems that need to be solved cooperatively.

3. Group cooperative learning is not based on independent thinking.

We should correctly handle the relationship between autonomous learning and cooperative learning. Before cooperative learning, let students think independently. After each student has a preliminary idea, they can explore and communicate together and solve problems together. This provides opportunities for students who don't like thinking or have certain difficulties in learning, which is helpful to improve their study. A hasty discussion without personal thoughts, such as passive water, expresses neither mature nor profound views, let alone individuality and originality. Organizing "group discussion" immediately after teachers ask questions often fails to achieve good results; In contrast, let students solve problems independently first, and then communicate with the whole class. When comparing various methods, they can be grouped according to different viewpoints, and then the whole class can communicate and debate as a unit, so that cooperative learning can receive good results.

4. The relationship between group cooperation and students' individual development has not been dealt with.

In the course of class, some students study in groups, while others are just listeners. On the other hand, when you talk to me, everyone is scrambling to speak, and no one listens to anyone, just expressing their opinions, which leads to extremely chaotic classroom order. Cooperative learning has only formal group activities, but no substantive cooperation; Good students have more opportunities to participate, which often helps. Difficult students become listeners, only listening to or watching the operation or speech of a good student, without interaction between students; Students are unfriendly, do not listen, do not share, etc. How do teachers deal with the relationship between group cooperation and personal development? Each member has his own advantages. Giving full play to everyone's wisdom and making everyone enjoy it is far more beneficial than a so-called good student's "concentrated discussion".

How to solve this problem? One is to ask students to learn to listen. Listening means listening to other people's opinions, understanding the key points, difficulties, methods and ideas to solve problems, and analyzing whether one's own methods and ideas are consistent with one's own when listening, so as to improve and absorb them. The second is to learn to share. When other people's views are different from mine, I will reflect on whether his method is correct. When his method is better than mine, I will absorb it for my use, which will share other people's thinking methods and learning results. This is just as Bernard Shaw said: "If two people have an apple, it is still an apple for each other. If two people have an idea, it will become two ideas for each other. " Similarly, in group cooperative learning, if everyone in a four-person group has different methods, then everyone can learn four methods and choose the best one. The third is to strengthen the incentive mechanism. For underachievers, we should open their talk boxes, give them more opportunities to answer with simple questions, and then inspire them with positive, affirmative and encouraging words, so that they can build up confidence and speak boldly.

5. Teachers failed to standardize group cooperative learning activities in time.

In the process of students' cooperative learning, teachers should patrol among groups, guide students to solve various problems in the learning process in time, help students improve their cooperative skills, and pay attention to observing students' performance in learning and interpersonal relationships, so as to be aware of them. Let students with certain learning difficulties think more and talk more, and ensure that they meet the basic requirements; At the same time, we should also give students who have the spare capacity to study a chance to develop their potential. In the process of group activities, teachers should strengthen the timely regulation of each group, pay special attention to the performance of students with learning difficulties in the activities, and give them more opportunities to speak.

However, at present, the improper regulation of cooperative learning teachers has the following manifestations:

1. Underestimate students' ability to understand knowledge and take their own understanding as the standard to replace students' understanding. If the teacher asks questions, ask the students after a few minutes of cooperative study. When no one answers, the teacher thinks that no students understand, and as a result, he replaces the students' understanding with his own explanation.

2. Prompt the key points, difficulties and contradictions of the problem prematurely, so that students' understanding of contradictions still stays at the low level of sensibility. For example, in the teaching of "rectangular area calculation", the teacher took out a piece of cardboard with a length of 5 cm and a width of 3 cm, and deduced that its area was 5 ×3= 15 square cm. Then the teacher asked: What is the relationship between the area of a rectangle and its length and width? Teachers do not solve this problem by asking students to discuss and cooperate with each other, but by asking and answering questions. This makes students' understanding of this problem only stay at a low level.

Of course, there are other misunderstandings between inquiry learning and group cooperative learning. In teaching, we should constantly study countermeasures to light up the red light for these misunderstandings. Make inquiry learning and group cooperative learning really play the main role of students, inspire each other through group cooperative discussion, realize complementary advantages and solve problems that individuals can't solve; Give full play to each student's enthusiasm and creativity in solving problems, cultivate students' team spirit, tap the potential of individual learning, and make students complement each other.