20 17 Excellent Papers of the National Mathematical Modeling Competition for College Students 1 Analysis of Mathematical Modeling Course Reform and Teaching Methods
Paper Keywords: Mathematics curriculum; Mathematical modeling; Curriculum setting; Curriculum revolution
Abstract: Developing mathematical modeling teaching and competition is an important way to cultivate students' innovative ability. This paper analyzes the problems existing in mathematical modeling competition, and finds that the root of the problems is related to the unreasonable setting of compulsory courses and specialized courses. We should reform the setting and teaching methods of mathematics courses in colleges and universities, and put forward specific reform suggestions.
1. Introduction
Mathematical modeling, from a macro point of view, is a process in which people transform and conquer nature with the help of mathematics, and from a micro point of view, it is a teaching activity mode in which mathematics is used as a tool to solve practical problems. Mathematical modeling education itself is a kind of quality education. Mathematical modeling teaching and competition is an effective way to implement quality education, which not only enhances students' awareness of mathematical application, but also improves students' ability to analyze and solve problems by using mathematical knowledge and computer technology. Therefore, strengthening mathematical modeling education and cultivating students' awareness and ability of mathematical application have become one of the important goals of mathematical modeling curriculum reform in colleges and universities in China. Although many colleges and universities in China have made some achievements in mathematical modeling, many problems have also been exposed in the competition, which has led to thinking about traditional courses and teaching methods.
2. The present situation, existing problems and reasons of mathematical modeling are analyzed.
2. 1 model contest status
According to the competition time (mid-to-late September), most colleges and universities in China generally start organizing students to register and train in mid-July every year. The training content is divided into two parts: first, it focuses on explaining some basic knowledge, mainly including ordinary differential equations, probability and mathematical statistics, operational research, mathematical experiments, modeling basics and other courses; Then, carry out modeling and simulation training. Some of the competition questions in previous domestic and foreign general groups and junior college groups are multiple-choice questions, so that students can voluntarily divide into groups and complete them within the specified time, and voluntarily explain their own ideas and methods of solving problems.
Students who take part in the competition must take part in training first. They usually pay attention to the notices on the campus network first, and then sign up voluntarily in various departments. After the training, the contestants are selected. In fact, there is generally no selection process for students participating in the competition. Basically, the number of trainees automatically decreases during the training stage, and the rest is the number of trainees. In recent years, the composition of students participating in training and competitions is basically similar. Not many students signed up, most of them came to see what was going on. After listening to one or two classes, you will automatically disappear or quit.
The teaching content of mathematical modeling course is problem-centered and arranged in blocks; The course of mathematical modeling has been offered for a short time, and there is a lack of teaching experience for reference. Most teachers use the mechanical explanation of the model. As for the background of the problem, many mathematical ideas and methods that may be used in the process of modeling are rarely taken into account, let alone let students discuss, communicate and cooperate in class, which makes it difficult for students to master the ideas and methods of mathematical modeling.
2.2 Problems and causes analysis
As can be seen from the above, there are some problems in the modeling of most universities in China. First, the work of mathematical modeling is not included in the daily teaching work, and students are not interested in mathematical modeling and their enthusiasm is not high. Secondly, the students participating in the training competition have a single major, and the mathematical modeling activities are not comprehensive. Although this is related to publicity, it is mainly due to the lack of necessary teaching links. Third, there are fewer senior students participating in the competition, but a large proportion of them win prizes. Senior students, in particular, are under great pressure from graduation, employment and postgraduate entrance examination. Although they have a deep foundation in mathematics, they have no intention of considering competition. There are a large number of junior students who take part in the training competition, and they are highly motivated, but they can't get any results. This shows that mathematical modeling is closely related to the mastery and accumulation of knowledge, and it is a process of combining theory with practical application, integrating and releasing knowledge. The curriculum of lower grades is unreasonable, and some related courses are offered too late. Fourth, many people think that courses should focus on solving practical problems with complex backgrounds. People who hold this view mainly ignore the characteristics and training objectives of mathematics education specialty. In our opinion, the focus of mathematical modeling course for mathematics education major should be to establish faith, cultivate consciousness and ability.
In addition, there are many shortcomings in the setting of mathematical modeling courses and the use of teaching materials. It is understood that most of the teaching modeling courses for mathematics education majors in colleges and universities copy the mathematics modeling textbooks for science and engineering majors. These textbooks mainly have the following problems: first, the textbooks mainly cover a large number of difficult ready-made mathematical models, which apply a lot of knowledge and methods in non-mathematical fields. To understand these problems, students majoring in mathematics education lack the proper foundation and can only learn by imitation and mechanical memory; Second, the textbook mainly adopts the block arrangement system with the problem as the main line, with the list of problems as the main part, and excessively highlights the problem solving. Copying this kind of teaching material has brought a great negative impact on the teaching of mathematical modeling for mathematics education majors, which is difficult for students to accept and teachers to control. More importantly, it is difficult to implement the course of mathematical modeling for mathematics education majors. Mathematics is widely used? It is difficult to meet the needs of the reform of mathematics education in colleges and universities to cultivate students' awareness and ability of mathematics application and make them master a set of mathematical modeling methods.
To sum up, we believe that it is the most important thing to solve the problems existing in setting up mathematical modeling courses for mathematics education majors, and it is imperative to construct teaching materials that conform to the reality and characteristics of mathematics education majors and form a set of scientific teaching methods that adapt to the characteristics of mathematics education majors.
3. Ways and methods to carry out mathematical modeling teaching with mathematical modeling activities as the carrier.
At present, there are many ways and methods to carry out mathematical modeling teaching, among which the most common and effective way and method is to carry out mathematical modeling teaching with mathematical modeling activities as the carrier, and its ways and methods can be described as follows:
3. 1 Carefully design teaching cases and implement case teaching method.
The so-called case teaching method is that in classroom teaching, teachers take specific cases as the main teaching content and introduce modeling thinking methods through modeling examples of specific problems. Part of the classroom activity is the teacher's lecture and part is the classroom discussion, that is, students speak, put forward their own understanding of the problem and the established mathematical model, and put forward a new mathematical model for solving, analyzing and discussing, and make a comparative test. To implement case teaching, we should grasp the following links:
(1) Teaching case selection. To make case teaching achieve the best effect, the most important thing is to choose a good teaching case. The following principles should be followed when selecting cases: ① representativeness. Cases should avoid involving too much professional knowledge, but also consider the relationship between scientific development and disciplines, and broaden students' knowledge. ② Primitive. Reports, plans, statistics and other information from radio and television, newspapers and magazines, government agencies, enterprises and institutions. Are important raw materials for mathematical modeling problems; You can also guide students to personally go to the front line to investigate and study, and pay attention to accumulating subject materials. 3 fun. When selecting specific cases, we should choose cases that are both interesting and fully embody the idea of mathematical modeling, such as population problem, seven bridges problem, Jinluo: Wolf team crossing the river problem, three-stage rocket launching satellite problem, forest fire fighting problem and so on. Starting with cultivating interest, let students gradually realize the thinking method and importance of modeling. 4 innovation. When writing modeling examples, we must consider cultivating students' innovative spirit and creativity. Therefore, we should pay attention to the compilation of examples such as one topic with multiple modules or multiple topics with one module and statistical charts, and pay close attention to the development of modern science and technology, so that students' innovation can be closely combined with high technology and integrated into the mainstream of contemporary scientific development.
(2) Case classroom teaching. Teachers should pay attention to two aspects when teaching specific modeling cases. In the first aspect, we should start from the actual problem, explain the background of the problem, the requirements of modeling and the information we have, and how to establish an optimized mathematical model through reasonable assumptions and simplified analysis. It also emphasizes how to explain the actual phenomenon and test the model with the solution results. This method not only highlights the teaching focus, but also leaves room for students to think further. For example, in the teaching of infectious disease model, different assumptions will lead to the establishment of different models, and only by proceeding from reality and constantly revising can it become a successful model. In addition, some improvement directions can be provided for students to explore and learn independently after class. Another aspect is that the teacher's teaching must be combined with the students' discussion. When the teacher clearly explains the case background, key factors and mathematical tools used. What kind of mathematical knowledge and ideas should be used and what kind of mathematical model should be established so that students can express their opinions and carry out discussion teaching. On the one hand, this can avoid the teacher's? Full house irrigation? On the other hand, it can enliven the classroom atmosphere, improve students' interest and enthusiasm in classroom learning, turn imparting knowledge into learning knowledge and applying knowledge, and truly achieve the teaching purpose of improving quality and cultivating ability.
3.2 Do a good job in after-school modeling practice training to consolidate and deepen classroom teaching.
In order to consolidate and deepen classroom teaching content and further improve students' modeling ability, modeling practice training is also an important part of mathematical modeling teaching. There are mainly the following forms: First, arrange after-school training questions. The first kind of training questions can be modeling with the mathematical modeling method mentioned in class, or further discussing a problem in class, which is to achieve the purpose of consolidating classroom teaching.
The other is to achieve the purpose of deepening classroom teaching. After learning the mathematical knowledge unit, the students are assigned training questions about the knowledge unit, and the students are intensively trained in the mathematical modeling laboratory in a specific period of time. Every training question should be completed completely, from asking questions, analyzing problems, establishing models, solving models to analyzing, testing and popularizing models, and a mathematical paper with clear thinking and clear order should be completed within the specified time. Through intensive training in this process, students' ability to identify, model and use models has been fully exercised and improved. The first step after each training question is that the teacher carefully examines the training paper and puts forward correct opinions on the problems in the paper in time; The second link is to organize the whole class to hold a special discussion on the training paper, so that students can tell the ideas, modeling ideas and methods of the paper. Through overall communication, let everyone learn from each other, learn from each other's strengths, and achieve the goal of * * * promotion. The second is to teach math software systematically and let students practice on the computer. With the development of computer technology, some high-performance applied mathematical software came into being, such as Matlab, Mathematica, Mapple, SAS, Lindo, Lingo and so on. With the appearance of these mathematical softwares, the complicated data calculation and processing in textbooks is no longer a difficult problem. After systematically teaching the specific use skills of these mathematical softwares, the teachers let the students operate on the computer in person and master the application of these softwares in actual mathematical operations. For example, how to calculate derivatives, integrals and limits with software; How to use software to solve equations, equations and linear programming; How to use mathematical software to study the law of function change, draw curves and surfaces and so on.
3.3 Continuously improve the level of mathematics teachers and promote the teaching of mathematical modeling.
In the teaching of mathematical modeling, teachers are the key. The level of teachers directly determines whether the teaching of mathematical modeling can achieve the expected purpose of cultivating students' ability. Teachers who teach mathematical modeling not only require high professional level, but also have rich practical experience and strong ability to solve practical problems. Therefore, in order to improve the level of teachers, on the one hand, teachers can be sent out for professional training and academic exchanges, such as attending various academic conferences and visiting scholars at prestigious schools. On the other hand, more famous experts and professors can be invited to give academic reports on modeling, so that teachers and students can increase their knowledge, broaden their horizons and understand the new trends and trends at the forefront of scientific development. In addition, mathematics teachers must update their educational concepts and constantly accumulate and update their professional knowledge, including a wider range of humanities and scientific literacy. Only by constantly innovating and striving to improve their own quality can math teachers adapt to the new situation and meet the requirements of the times.
In a word, the content of mathematical modeling has practical value, the teaching of mathematical modeling course can be lively and interesting, and mathematical modeling can have products and achievements of knowledge innovation. In particular, to promote the teaching of related mathematics courses, mathematical modeling should be set up after students have finished studying related courses or in related courses, and at least some mathematical experiments should be arranged in the existing teaching content.
References:
Li Daqian. Mathematical modeling competition for Chinese college students [M]. Beijing: Higher Education Press, 1998.
[2] An Shuhua. Reflections on the reform of mathematics education in China [J]. Journal of Mathematics Education, 2004.
[3] Tai-Huang An. Mathematics teachers' view of mathematics and mathematics education [J]. Journal of Mathematics Education, 2004.
[4] Wang Maozhi. Discussion on the design of mathematical modeling training course system [J]. Journal of Mathematical Education, 2005.
20 17 the teaching of mathematical modeling ideas, the second excellent paper of the national college students' mathematical modeling competition
1 the significance of integrating mathematical modeling into the teaching of linear algebra
1. 1 Stimulate students' interest in learning and cultivate their innovative ability.
The essence of education is to enable students to apply what they have learned while mastering knowledge. However, at present, the teaching of linear algebra pays more attention to theory than application, so students feel boring in class and have poor learning initiative, let alone innovation. If teachers can integrate the ideas and methods of mathematical modeling into the daily teaching of linear algebra, they can not only stimulate students' interest in learning linear algebra, but also stimulate students' enthusiasm in using linear algebra knowledge to solve practical problems, so that students can realize the true value of linear algebra, thus changing the concept of uselessness of linear algebra and cultivating students' innovative ability.
1.2 Improve the attraction of linear algebra course and increase students' benefits.
Mathematical modeling is the best performance of cultivating students to use mathematical tools to solve practical problems. If the ideas and methods of mathematical modeling are infiltrated into the teaching of linear algebra, it can not only stimulate students' interest in learning linear algebra, but also make them understand that seemingly boring definitions and theorems are not passive water, but have realistic background and practical use, which can greatly improve the boring situation of linear algebra classroom and enhance the attraction of linear algebra course. Judging from the present situation of mathematical modeling teaching, students have not benefited much. But in any university, science, engineering and economic management majors will offer three compulsory courses: advanced mathematics, linear algebra and probability statistics. If the ideas and methods of mathematical modeling can be infiltrated into the teaching of compulsory courses such as linear algebra, advanced mathematics and probability statistics, the benefits of students will be greatly increased.
1.3 to promote the self-improvement of linear algebra teachers
In order to integrate the ideas and methods of mathematical modeling into the course of linear algebra, linear algebra teachers are required not only to have good theoretical knowledge teaching skills, but also to have the ability to use linear algebra knowledge to solve practical problems, which forces linear algebra teachers to constantly learn new knowledge and technology, promote the continuous updating of their own knowledge, and thus achieve the effect of improving teaching and scientific research capabilities.
2 into the teaching of mathematical modeling of linear algebra
Thinking Mode Although the course of Linear Algebra itself has many contents and insufficient class hours, it is useless for us to integrate the idea of mathematical modeling into the course of Linear Algebra. Mathematical modeling? The course content preempted the class of linear algebra. Here, the author will gradually infiltrate the idea of modeling into the teaching of linear algebra from the following two aspects.
2. 1 Integrate the idea of mathematical modeling into the concept of linear algebra.
Broadly speaking, complex and abstract concepts such as determinant, matrix, matrix multiplication, vector and linear equations in linear algebra textbooks all come from reality. Therefore, when teaching these concepts, we can properly select some vivid examples to attract students' attention, and at the same time, we can naturally establish conceptual models so that students can fully feel the transformation from practical problems to mathematics. For example, matrix is an important concept in linear algebra. When introducing the concept of matrix, we can start with a simple input-output problem and use a rectangular table to represent the data in this problem. This simplification idea well embodies the idea of modeling abstraction, and such a rectangular table is called a matrix.
2.2 Incorporate the idea of mathematical modeling into the homework of linear algebra.
Homework is the digestion and consolidation of classroom teaching content. But at present, the exercises in textbooks and related reference books do not involve the definitions and theorems in linear algebra. In order to make up for this, we can add some linear algebraic modeling problems to the exercises, as follows. 1) After learning 1~2 units, students will carry out 1 large-scale homework according to what they have learned, and students can complete the homework in groups of three (that is, 1 essay). In the process of completing homework, students can not only strengthen and consolidate the classroom teaching content of linear algebra, but also improve their self-study ability and thesis writing ability and cultivate team spirit. At the same time, students can get in touch with scientific research methods as soon as possible by completing large-scale homework, which is consistent with the current purpose of encouraging college students to carry out scientific research innovation. 2) After all the students' homework is completed, students can be organized to explain the ideas and problems encountered in completing the homework, while teachers make corresponding comments on different articles and point out the direction of improvement. This transposition teaching mode can not only urge students to finish their homework better, but also improve their language expression ability and promote the relationship between teachers and students, thus greatly improving the teaching effect.
3 in the teaching of linear algebra into mathematical modeling
Thought case case 1: input-output problem [4]. There are coal mines, power plants and railways somewhere. According to the cost accounting, every coal produced 1 yuan needs to consume electricity in 0.3 yuan; In order to transport 1 yuan coal out, it will cost 0.2 yuan freight; Every time 1 yuan of electricity is produced, 0.6 yuan coal is needed as fuel; In order to run the auxiliary equipment of the power plant, it needs to consume 0. 1 yuan for electricity and 0. 1 yuan for freight; As a railway administration, 0.5 yuan coal will be consumed for every transportation that provides 1 yuan freight, and auxiliary equipment will consume 0. 1 yuan electricity. Now the coal mine has received 60,000 yuan of foreign coal orders, and the power plant needs 654.38+10,000 yuan of electricity. Q: How much can coal mines and power plants produce to meet the demand? Model assumption: it is assumed that other factors such as price changes are not considered.
4 conclusion
It is feasible to integrate the mathematical modeling idea into the teaching of linear algebra and cultivate students' modeling ability, which meets the requirements of contemporary talent training. At the same time, we should also realize that the original main mathematics curriculum system is the product of years of historical accumulation and inspection, and it is not appropriate to completely change it without sufficient basis [6]. Therefore, the idea of mathematical modeling should be gradually combined with the existing teaching content as far as possible. Practice has proved that integrating mathematical modeling into linear algebra teaching not only stimulates students' interest in learning, cultivates students' innovative ability, but also promotes teachers' self-improvement. However, how to integrate the idea of mathematical modeling into the teaching of linear algebra is still in the exploratory stage, and it needs the joint efforts of the majority of mathematics teachers.
& gt& gt& gt More exciting next page? 20 17 outstanding papers of the national mathematical modeling competition for college students?
Guide reading:
On the evening of February 17, the "Touching China 2020 People Award Ceremony" was broadcast, and Zhang was elected as the person of the yea