Cooperative learning is a creative and effective teaching theory and strategy that arose in the United States in the early 1970s, and made substantial progress in the mid-1970s and mid-1980s. Because of its remarkable effect in improving teaching atmosphere, improving students' academic performance and promoting students to form good non-cognitive quality, it has quickly attracted the attention of all countries in the world and become one of the mainstream teaching theories and strategies in contemporary times. In the course of university, there is the content of cultivating cooperative learning, and the most prominent subject is mathematical modeling. China Mathematical Model Competition started at 1992. This competition is a real team competition. Each group consists of three people, who must complete an answer sheet within the specified three days. If you want to get good grades, you can't do it without hard training. At the beginning of mathematical model course training, we should first divide into groups, and the division of cooperative learning groups generally follows the following principles: 1. Grouping principle of cooperative learning 1, heterogeneous grouping principle, heterogeneous grouping, and the pursuit of interaction and cooperation between students. In the mathematical model, students are divided into several groups to study, which is not simply to let several students get together to study or discuss. Grouping should advocate the principle of "heterogeneous" grouping. The so-called "heterogeneous" grouping is to put 3~8 students with different grades, abilities, sexes and even different personalities in a cooperative group. In this way, the students in the group are different in ability, personality and gender, and they are complementary, which is convenient for students to learn from each other, help each other and give full play to the role of the group. Because each group is heterogeneous, it makes each group homogeneous, which lays the foundation for fair competition for each group on the same starting line. This is the grouping principle of cooperative learning, and it is also an important technology to realize "intra-group cooperation and inter-group competition" This grouping principle can be optimized at the end of training and into the intensive training stage, so that three players from each team can become the best partners, which is conducive to the improvement of competition results. 2. After defining the principle of responsibility, in order to prevent "responsibility spread", team members should not be scattered. Cooperative learning especially emphasizes the individual responsibility of each member in the group, so as to realize the benign interaction and cooperation among the members of the group, and make students realize that the members of the group are working together for the same goal. In order to complete the task faster and better, the members of the group must be interdependent, "honor and disgrace and * * *". In a cooperative team, the individual responsibilities of team members are usually defined by the distribution of roles and resources, so they are interdependent. Everyone in the group has his strongest one. Some people have strong expression skills, some people use computers well, and some people have solid knowledge of mathematics. It is impossible for everyone to understand and contact the problems in the mathematical model, so students who are skilled in computer operation can find relevant information on the Internet when they encounter problems, especially unfamiliar problems. Students with strong mathematical ability can be responsible for algorithm design, and then students with high programming level can write programs and display the results through computer simulation; Students who write well are "writers". Sometimes, other methods can be used to clarify students' responsibilities and achieve positive interdependence. For example, the total task is divided into subtasks and assigned to each member, and the completion quality of the total task is evaluated by the completion quality of subtasks. All these methods make the team members indispensable in the group, have their own clear responsibilities and must depend on each other, which also reflects the value of everyone in the mathematical model group. Second, there are two main problems in cooperative learning: 1. Cooperative learning is a mere formality, and the division of labor is unreasonable. Some students' reaction to cooperative learning is that they don't know how to study together effectively. It is relatively easy for several students to cooperate together to complete simple learning tasks; When the difficulty of the topic deepens, students are often at a loss and it is difficult to achieve a reasonable division of labor. To overcome this phenomenon, teachers' guidance is the key, pointing out the crux of the problem, and then strengthening the running-in step by step through the research and analysis of team members. 2. Lack of equal spirit of cooperation in cooperation activities. In the process of cooperation, some students may think they are capable and unwilling to accept different opinions, while others pretend to be bystanders. In this way, conflicts and disputes will inevitably arise in cooperation, and even there will be a phenomenon that everyone strives for merit. In view of this situation, on the one hand, we should attach importance to students' emotional communication and exchange, on the other hand, we should set an example and guide them reasonably. In the process of cooperative learning, learners can observe other people's behaviors and their results, summarize or comprehend the characteristics of other people's behaviors, form rules, and form their own behaviors by reorganizing these rules. Therefore, in-depth cooperation between students must also enable them to learn from each other, encourage each other and promote each other. In this cooperative atmosphere, the spark of innovation can often come up with unexpected answers. The evaluation concept of cooperative learning is also very different from traditional teaching. Changing the competition between individuals into the competition among groups, changing individuals into group scores, and taking the overall score of the group as the basis for reward or commendation, has formed a new pattern of "cooperation among group members and competition among group members", which has shifted the focus of the whole evaluation from encouraging individual competition to reaching the standard. Thirdly, the effective use of evaluation mechanism is the key to the effectiveness of cooperative learning. From the perspective of evaluation methods, cooperative learning includes individual evaluation and group evaluation, self-evaluation and peer evaluation, student evaluation and teacher evaluation. These groups of evaluations are mainly the former, but they can also be combined in many ways. Among them, group self-evaluation is very important. It is a reflection on which activities of group members are beneficial and which activities need to be improved in a certain period of group activities, with the aim of improving the effectiveness of the group in achieving the same goal. The evaluation methods of cooperative learning can also be divided into process evaluation and result evaluation. Among them, the process evaluation is the main one, which mainly evaluates the behavior, enthusiasm and participation of students in group cooperation, as well as the changes of students' emotions, attitudes and abilities in activities. Through the above multiple evaluations, we can identify and evaluate students' participation behaviors and effects, and promote mutual learning among students. It can guide students to continue to explore and learn in cooperation and make continuous progress in "collision, docking and integration"; Can make the appraisee get encouragement and spiritual support, and make him play a greater creative potential and enthusiasm for cooperation. Through this evaluation system, those students who think they are "hard-working" will soon understand that others' contribution is so great that others' contribution is no less than their own! Through the evaluation, I deeply realize the importance of the team, and the strength of the individual is negligible. Recognize that in the process of cooperative learning, when encountering differences, we should respect other people's opinions and speak our own opinions with an open mind. Learn from each other's strengths. According to personal specialties, clear division of labor, open-minded listening and reasonable collocation. The quantitative results of evaluation are divided into two parts: "basic score" and "improved score". The basic score is the evaluation score of the degree to which the student group completes the task; On the basis of intra-group evaluation and process evaluation, improving scores is the embodiment of students' contribution to the group. The purpose of introducing basic scores and improving scores is to give all students the opportunity to strive for the highest score for their group as much as possible, and to guide students to concentrate on striving for continuous progress and improvement. Cooperation promotes students' active participation in learning. Paying attention to cooperative behavior reduces students' "self-centeredness" and improves students' sense of responsibility for their own learning. There is evidence that compared with the classroom organization of individualized learning, cooperative learning has a lasting impact on students' academic performance, especially on social learning and personal self-esteem. In a word, cooperative learning pursues cooperation and communication between people, democracy and equality, harmony, mutual trust, active participation and improvement. More importantly, we live in an information age that requires cooperative awareness and social skills, especially today when the only child accounts for a considerable proportion. Cooperative learning integrates cooperation, competition and individual behavior, which conforms to the teaching law and the needs of the times. Group cooperative learning is in the exploratory stage, and the existing problems are not surprising. In education and teaching, how to make substantial progress in group cooperative learning is still a long-term and arduous task! [References] [1] Liu Jilin, Wang Tan. The basic idea of cooperative learning [J]. People's Education 2004.6438+0. [2] (America) Bruce Joyce Mashawell Emily calhoun. Teaching mode [M]. Beijing: China Light Industry Press. Cooperative learning theory in mathematical modeling

Abstract: Cooperative learning is a creative and effective teaching theory and strategy system widely adopted in many countries in the world today. This paper discusses the grouping standard, key system and basic theory of cooperative learning in mathematical modeling. Keywords: cooperative learning; Key systems; Beginners' Course of Mathematical Modeling Competition (IV)-Actual Combat-3D Reconstruction of Blood Vessels.

At the end of 200 1 September, we finally ushered in the national mathematical modeling competition for college students. At that time, there was a game just released in Xishanju. There is a song called the ruins of love: whose body is blue sky, plundered by clouds, leaving emotional evidence. When the feelings are slowly distorted in your heart, is my love wronged you, adding fear and sadness, and letting the stars clean it up? This irresponsible ending, who tore the sky from the wounds of the stars, obliterated my freedom and romantic tenderness. If the rain outside is the tears shed by the stars for me, I don't know if you have any injuries in your heart. If the rain in my heart comes from a broken house, I wonder if the memories I care about will become the ruins of buried love. I don't know why, I prefer the feeling of sadness, just like this song gives people the feeling that it is easy to produce strength. The day before the game, my two partners and I started to carry the necessary combat weapons into the hostel. Make a list: > a set of mathematics manuals (5 volumes, each as thick as a fist): advanced mathematics textbooks (published by Dongdian Military Academy), probability theory (Fudan University), numerical analysis (Dongdian Military Academy), some reference books of Matlab, C language courses (from Tan Haoqiang), etc. > My three computers are Celeron 533 and 566, and they all have network cards and UPS uninterrupted power supply. The software includes Matlab, VisualC++, Microsoft Word and Windows2000 operating system (I didn't learn Visio at that time, Other software seems to be nothing) > borrowed an HP Lasier Jet6.0 printer from the institute >: ......................................................................................................................... ......................................................... as soon as we were ready, we checked into the hostel outside the south gate of the school (the conditions were average before, although it was upgraded, haha, most people couldn't live). The teacher asked us to download the topic from the internet at eight o'clock the next morning, but I don't know who sent a message saying that the topic might be downloaded from the internet at night. So we didn't sleep well all night, so we went online from time to time to see if we could download the game questions. But I didn't get the title until eight o'clock the next morning. ) Mathematical modeling competition generally has three topics, two for undergraduates and two for specialists. One problem is that the specialist group is the same as the undergraduate group. The theme is: three-dimensional reconstruction of blood vessels and bus scheduling. You should choose one of these two questions to do. Which one should I choose? After careful study, we found that bus scheduling is an optimization problem, and the three-dimensional reconstruction of blood vessels focuses on algorithm. So the three of us did not hesitate to choose three-dimensional reconstruction of blood vessels. By the way, what is the reason? Because we did an optimization problem a year ago, it was a steel pipe transportation problem, and it was badly done, so everyone was worried and tried not to choose this topic, three-dimensional reconstruction of blood vessels. The first difficulty is-how to read these bmp images and save them as a binary matrix? At first, we went to the library to find the book bmp file format, and prepared to read bmp with C program. When we were just about to do it, we were surprised to find that there was a ready-made function imread in Matlab that could do it! God help me, too. I immediately read all the bmp images of 100, and converted the bmp file of each slice into 0, 1 matrix of 512. And use the save function, turn on the ASCII switch, and save each matrix as a txt file. So you can use the C program directly. In the above process, we found that the names of bmp given in the title are not very good, which range from 0, 1, 2 ... to 99, so we changed these names to 0 1, 02, 03, 04, ... 99, and changed all the file names to two digits for easy operation.

The next step is how to get the result. First of all, we searched the library for a long time to see if there were any papers that solved similar problems. Not only Chinese, but also English. By the way, English is really important. On the Internet, English is a well-deserved overlord. If you want to find information on the Internet, you can't do it without good English. We found that medical CT imaging technology can be used for reference. These materials are not necessarily useful, but they can open our minds well and it is worthwhile to spend time on them.

Then, we thought and thought, and thought and thought all the time. We used ACDSEE to hang the bmp image of 100 upside down like a slide, and also used something like skin to simulate blood vessels and make them bend around. Finally, guess intuitively-the radius of the circle with the largest radius that can be sliced is equal to the radius of the original ball (the ball that forms the envelope).

So we began to work separately, on the one hand, one person to prove this conclusion. On the other hand, start programming to realize this idea. In the process of writing the program, we also extended two assumptions: the radius of the circle that can be sliced must be less than or equal to the radius of the original ball; The radius of a circle that cannot be included in a slice must be greater than the radius of the original ball. Hehe, using these two assumptions, dichotomy can easily solve this program. But the program doesn't run well. We divide the program into three computers, and each computer calculates a part of graphics, which is also a parallel algorithm (parallel algorithm is the specialty of TEPCO Military Academy). Even so, it took a night. In the meantime, we also made some modifications to the algorithm and recalculated it. Indeed, the algorithm needs continuous improvement. Please look at this sentence: [Because of the limited precision of the given data, the radius of the original sphere contained in the slice may be more than one circle], which was discovered during the implementation of the algorithm. At first, it was hard to think of these details.

Another detail is that it is difficult to write this program with Windows console program or Dos program (turbo c). Because we need to use the minimum matrix of 5 12*5 12, we will use a larger matrix in the process of algorithm writing for convenience. But Dos doesn't support such a large matrix array, so I suggest you write a 32-bit Windows program.

We put forward these assumptions, and it is not easy to prove them completely scientifically. Sometimes, what he takes for granted, I think it should be proved; I think the proof of this logical confusion was confirmed by him as completely correct. Hehe, so, let's argue for a while, prove for a while, communicate for a while, and then argue. Once, I argued furiously, and my heart seemed to explode. I think, I won't do this competition! I am going back to school! Why should I cooperate with you? Why should I accommodate you? I quit! I choked back, didn't speak, went to the window, looked up at the blue sky outside, and suddenly remembered the song-"Whose body is the blue sky ...", and I slowly hummed, and in an instant, everything was quiet. I silently sat down beside the computer and continued programming. ...

I slept for 4 hours on the first night and 2 hours that night. Forget it, there is only one day left. On the third night, I didn't sleep because I had to catch up on my thesis.

Because we can't use Word, the numbering and typesetting of charts are purely manual, which is too bitter. Only when we are in it can we understand it. After a lot of physical labor, the paper was completed, and it was too late to check it carefully, so it was typed and handed in. As soon as we handed it in, we found that the drawing number was wrong. Alas, please remember our lesson!

By the way, a little experience in doing mathematical modeling problems. 1. Write down your hypothesis at any time. Sometimes we will start the next step under our own reasonable assumptions, so we should write this assumption down conveniently, or we will forget it in the end. But also make our answers more rigorous. 2. Record your thoughts at any time and express them completely without leaving any room. After the competition, many students often complain when the teacher comments on excellent papers. I also thought of this idea, but I didn't express it or express it clearly. But it is often this thing that others have not clearly expressed that has contributed to an excellent paper. 3. It has its own characteristics. There are so many papers in mathematical modeling contest, why should the teacher vote for himself? Of course, it must have its own characteristics. Being popular means having your own bright spot.