Mathematics, modeling, teaching, leadership

When it is necessary to analyze and study a practical problem from a quantitative point of view, people should, on the basis of in-depth investigation and study, understanding the object information, making simplified assumptions and analyzing the internal laws, express it as a mathematical formula, that is, a mathematical model, with mathematical symbols and language, and then use the calculated model results to explain the actual problem and accept the actual test. This whole process of establishing mathematical model is called mathematical modeling.

Over the past half century, with the rapid development of computer technology, the application of mathematics has not only played an increasingly important role in engineering technology, natural science and other fields, but also penetrated into new fields such as economy, finance, biology, medicine, environment, geology, population and transportation with unprecedented breadth and depth. The so-called mathematical technology has become an important part of modern high technology.

mathematical modeling

No matter how to solve practical problems in the fields of science, technology and production by mathematical methods, or combine with other disciplines to form an interdisciplinary subject, the first and key step is to establish a mathematical model of the research object and calculate and solve it. Mathematical modeling and computer technology play a more powerful role in the era of knowledge economy.

Mathematical modeling application

Mathematics is a science that studies quantitative relations and spatial forms in the real world. In the long history of its emergence and development, it has always been closely related to various application problems. The characteristics of mathematics lie not only in the abstraction of concepts, the rigor of logic, the clarity of conclusions and the integrity of systems, but also in the universality of applications. Since the 20th century, with the rapid development of science and technology and the increasing popularity of computers, people's requirements for various problems have become more and more precise, making the application of mathematics more and more extensive and in-depth. Especially in the era of knowledge economy that is about to enter 2 1 century, the position of mathematical science will change greatly, and it is changing from. With the globalization of economic development, the rapid development of computers and the continuous expansion of mathematical theories and methods, mathematics has become an important part and think tank of contemporary high technology and a technology that can be universally implemented. Cultivating students' consciousness and ability of applying mathematics has become an important aspect of mathematics teaching.

The significance of editing this mathematical modeling

mathematical modeling

Mathematical modeling is a mathematical thinking method, and it is a powerful mathematical means to describe and "solve" practical problems by using mathematical language and methods through abstraction and simplification. Mathematical modeling is a process of describing actual phenomena with mathematical language. The actual phenomena here include both concrete natural phenomena, such as free fall, and abstract phenomena, such as customers' value tendency to a certain commodity. The description here includes not only the description of external form and internal mechanism, but also the prediction, experiment and explanation of actual phenomena. We can also intuitively understand this concept: mathematical modeling is a process that makes pure mathematicians (mathematicians who only know mathematics but don't know its application in practice) become physicists, biologists, economists and even psychologists. A mathematical model is generally a mathematical simplification of actual things. It often exists in an abstract form close to the real thing in a sense, but it is essentially different from the real thing. There are many ways to describe an actual phenomenon, such as recording, video recording, metaphor, rumors and so on. In order to make the description more scientific, reasonable, objective and repeatable, people use a generally accepted strict language to describe various phenomena, which is mathematics. What is described in mathematical language is called a mathematical model. Sometimes we need to do some experiments, but these experiments often use abstract mathematical models as substitutes for actual objects and carry out corresponding experiments. The experiment itself is also a theoretical substitute for the actual operation.

Applied mathematical model

When applying mathematics to solve various practical problems, it is a very crucial and difficult step to establish a mathematical model. The process of establishing teaching model is the process of simplifying and abstracting complex practical problems into reasonable mathematical structures. Through investigation and data collection, observe and study the internal characteristics and laws of actual objects, grasp the main contradictions of problems, establish quantitative relations reflecting actual problems, and then analyze and solve problems by using mathematical theories and methods. This requires a deep and solid mathematical foundation, keen insight and imagination, strong interest in practical problems and extensive knowledge. Mathematical modeling is the bridge between mathematics and practical problems, the medium through which mathematics is widely used in various weapons, and the main way to transform mathematical science and technology. The important role of mathematical modeling in the development of science and technology has been paid more and more attention by the mathematical and engineering circles, and it has become one of the necessary abilities of modern scientific and technological workers. In order to meet the needs of scientific and technological development and cultivate high-quality and high-level scientific and technological talents, mathematical modeling has been gradually carried out in university education. More and more colleges and universities at home and abroad are offering mathematical modeling courses and participating in open mathematical modeling competitions, which are regarded as an important aspect of teaching reform and training high-level scientific and technological talents in colleges and universities. Now many colleges and universities are combining mathematical modeling with teaching reform. Strive to explore more effective teaching methods of mathematical modeling, and provide new ideas for cultivating talents facing 2 1 century. Compared with other mathematics courses in colleges and universities in China, mathematical modeling has the characteristics of great difficulty, wide coverage, flexible form and high requirements for teachers and students. Mathematical modeling teaching itself is a process of continuous exploration, innovation, improvement and improvement. In order to change the traditional teacher-centered, classroom-based and knowledge-imparting teaching mode, the guiding ideology of mathematical modeling course is: organizing teaching work based on laboratory, taking students as the center, taking problems as the main line and aiming at cultivating ability. Through teaching, let students understand the whole process of analyzing and solving problems by using mathematical theories and methods, and improve their ability to analyze and solve problems; Improve their interest in learning mathematics and their awareness and ability to apply mathematics, so that they can often think of using mathematics to solve problems in their future work, improve their awareness of making full use of computer software and contemporary high-tech achievements, and organically combine mathematics with computers to solve practical problems. The mathematical model is oriented to students. Teachers use some pre-designed questions to inspire students to actively consult documents and learn new knowledge, encourage students to actively carry out discussions and debates, cultivate students' active exploration and enterprising style of study, cultivate students' initial ability to engage in scientific research, cultivate students' spirit of unity and cooperation, and form a lively environment and atmosphere. The focus of the teaching process is to create an environment, induce students' desire for learning, cultivate students' self-study ability, enhance students' mathematical quality and innovation ability, and improve students' mathematical quality. Emphasis is placed on the ability to acquire new knowledge and the process of solving problems, rather than knowledge and results. Most students who take part in the pre-competition training of mathematical modeling contest need to learn "short courses" (or lectures) such as mathematical statistics, optimization, graph theory, differential equations, calculation methods, neural networks, analytic hierarchy process, fuzzy mathematics and the use of mathematical software packages. And most of them are enlightening to talk about some basic concepts and methods, mainly relying on students themselves to fully mobilize their enthusiasm and give play to their potential. Seminar is widely used in training, and students can report, discuss and debate by themselves. Teachers mainly play the role of asking questions, answering questions and counseling. The competition must use computers and corresponding software, such as Spss, Lingo, Mapple, Mathematica, Matlab and even typesetting software.

Edit this process

Model preparation

Understand the actual background of the problem, clarify its practical significance, and master all kinds of information of the object. Describe the problem in mathematical language.

Model hypothesis

According to the characteristics of the actual object and the purpose of modeling, the problem is simplified with accurate language and some appropriate assumptions are put forward.

Model structure

On the basis of hypothesis, use appropriate mathematical tools to describe the mathematical relationship between variables and establish the corresponding mathematical structure (try to use simple mathematical tools).

Model solving

Using the obtained data, all parameters of the model are calculated (or approximately calculated).

model analysis

The results are analyzed mathematically.

model testing

The model analysis results are compared with the actual situation to verify the accuracy, rationality and applicability of the model. If the model is in good agreement with the actual situation, the practical significance of the calculation results should be given and explained. If the model is not consistent with the actual situation, it is necessary to modify the assumptions and repeat the modeling process.

Model application

The application pattern varies with the nature of the problem and the purpose of modeling.

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Enter universities in western countries.

Mathematical modeling was introduced to some universities in western countries in the 1960s and 1970s, and several universities in China also introduced mathematical modeling into classrooms in the early 1980s. After more than 20 years' development, most undergraduate colleges and many junior colleges have offered various courses and lectures on mathematical modeling, which has opened up an effective way to cultivate students' ability to analyze and solve practical problems by using mathematical methods. The mathematical modeling competition for college students first appeared in the United States in 1985. 1989, under the organization and promotion of several teachers engaged in mathematical modeling education, students from several domestic universities began to participate in American competitions, and their enthusiasm became higher and higher. In recent years, the number of participating schools and teams has accounted for a considerable proportion. It can be said that the Mathematical Modeling Competition was born in the United States, blossomed in China and achieved fruitful results.

in China

1992, the Chinese society of industrial and applied mathematics organized a mathematical modeling league for college students in the national city of 10, with 3 14 teams from 74 universities participating. The leaders of the Ministry of Education discovered, fostered and cultivated this new thing in time, and decided that the Ministry of Education of People's Republic of China (PRC) and the Chinese Society of Industrial and Applied Mathematics would jointly host a national mathematical modeling competition for college students every year starting from 1994. In the past ten years, the scale of this competition has increased at an average annual rate of more than 25%. In 2009, there were 1 137 colleges and universities and 15046 teams (including team A 12276 and team B 2770) in 33 provinces/municipalities/autonomous regions (including Hong Kong and Macao Special Administrative Regions) in China, with 4500 college students of various majors.

Edit this mathematical modeling contest for college students.

contemporary undergraduate mathematical contest in modeling

The National Mathematical Modeling Competition for College Students is a mass scientific and technological activity for college students nationwide, which is co-sponsored by the Ministry of Education of People's Republic of China (PRC) and the Chinese Society of Industrial and Applied Mathematics. Its purpose is to stimulate students' enthusiasm for learning mathematics, improve their comprehensive ability to use computer technology to build mathematical models and solve practical problems, encourage students to actively participate in extracurricular scientific and technological activities, broaden their knowledge, cultivate innovative spirit and cooperative consciousness, and promote the reform of college mathematics teaching system, teaching content and teaching methods. Competition topics generally come from practical problems that have been properly simplified in engineering technology and management science. Participants are not required to master in-depth professional knowledge in advance, but only need to have studied mathematics courses in ordinary universities. The topic has great flexibility, allowing participants to exert their creative ability. Participants should complete a paper (that is, an answer sheet) including the hypothesis of the model, the establishment and solution, the design and computer realization of the calculation method, the analysis and test of the results, and the improvement of the model. Competition awards are based on the rationality of assumptions, the creativity of modeling, the correctness of results and the clarity of text expression. The national unified competition topic, adopts the communication competition way, carries on in the relatively centralized form; The competition is usually held within three days at the end of September each year; College students participate in the competition in teams of three, regardless of major.

Articles of Association of the National Mathematical Modeling Competition for College Students (2008)

Article 1 General Provisions The National Mathematical Modeling Competition for College Students (hereinafter referred to as the Competition) is a mass scientific and technological activity for college students all over the country jointly sponsored by the Higher Education Department of the Ministry of Education and the China Society of Industrial and Applied Mathematics. Its purpose is to stimulate students' enthusiasm for learning mathematics, improve their comprehensive ability to use computer technology to build mathematical models and solve practical problems, encourage students to actively participate in extracurricular scientific and technological activities, broaden their knowledge, cultivate innovative spirit and cooperative consciousness, and promote the reform of college mathematics teaching system, teaching content and teaching methods. Article 2 Competition Content Competition topics generally come from practical problems that have been properly simplified and handled in engineering technology and management science. Participants are not required to master in-depth professional knowledge in advance, but only need to have studied mathematics courses in colleges and universities. The topic has great flexibility, allowing participants to exert their creative ability. Participants should complete a paper (that is, an answer sheet) including the hypothesis of the model, the establishment and solution, the design and computer realization of the calculation method, the analysis and test of the results, and the improvement of the model. Competition awards are based on the rationality of assumptions, the creativity of modeling, the correctness of results and the clarity of text expression. Article 3 Competition Form, Rules and Discipline 1. The topics of the national unified competition are relatively concentrated in the form of communication competition. The competition is held once a year, usually within three days around a weekend. 3. College students participate in the competition as a team, with 3 people in each team (they must belong to the same school), and their majors are not limited. The competition is divided into undergraduate and specialist groups. Undergraduate students participate in undergraduate group competitions, junior college students participate in specialist group competitions (or participate in undergraduate group competitions), and graduate students are not allowed to participate. Each team may have an instructor (or teacher group) who is engaged in pre-competition counseling and organizing competitions. However, during the competition, the contestants must avoid, and must not guide or discuss, otherwise it will be regarded as a violation of discipline. 4. During the competition, the team members can use all kinds of books, computers and software to browse online, but they are not allowed to discuss with anyone outside the team (including online). 5. After the competition starts, the competition topics will be posted on the designated website for each team to download, and each team will complete the answers within the specified time and hand in the papers on time. 6. Participating universities should instruct relevant functional departments to be responsible for the organization and discipline supervision of the competition to ensure the standardization and fairness of the competition. Article 4 Organizational form: 1. The contest is presided over by the Organizing Committee of the National Mathematical Modeling Competition for College Students (hereinafter referred to as the National Organizing Committee), which is responsible for starting registration, drafting contest questions, organizing the evaluation and award of national excellent answer sheets, printing award certificates, and holding a national award ceremony every year. The competition will be organized by the department. In principle, a province (autonomous region or municipality directly under the Central Government) is regarded as a competition area, and at least 20 teams from 6 universities should participate in each competition area. Adjacent provinces can be merged to form a competition area. Organizing committees (hereinafter referred to as organizing committees) shall be established in each competition area to be responsible for the publicity, registration, supervision and marking of competition disciplines. Provincial university teams that have not set up a competition area can directly register for the competition with the National Organizing Committee. 3. Establish an organizational excellence award to commend the organizing committee of the competition area that has made outstanding achievements or remarkable progress in the organization of the competition. The main criteria are the number of participating schools and teams, the number and quality of questions, no violation of discipline, the quality of marking work, creative work combined with the specific conditions of the competition area, and cooperation with the National Organizing Committee.

The application of mathematical modeling is a great promotion to the mathematical modeling competition. At present, Beijing Noah Mathematical Modeling Technology Co., Ltd., the first mathematical modeling company in China, was established in Beijing. Wei Yongsheng, who has been studying for a doctorate, cooperated with two other like-minded students to start a business project, which originated from their familiar field of mathematical modeling. Wei Yongsheng set up a mathematical modeling competition team for college students in April 2003, and won the national second prize that year. In 2005, he won the first prize in the international mathematical modeling competition, and registered the website of mathematical modeling enthusiasts in June 2005. In line with the direction of mathematical modeling towards society and application, they formally established the entrepreneurial direction of mathematical modeling application in June last year, set up an entrepreneurial team and started the road of entrepreneurship. At the beginning of this month, Beijing Noah Mathematical Modeling Technology Co., Ltd. was officially registered, and the entrepreneurship of Wei Yongsheng's team was officially on the right track. At present, Noah Mathematical Modeling is expanding its business and strength from a professional perspective, and is actively involved in mathematical modeling, mathematical model solutions and consulting services in other related fields such as railway transportation, highway transportation and logistics management. Wei Yongsheng explained to reporters that many people may not understand the use of mathematical modeling. He cited an example, a railway station, to calculate how often the train leaves, to ensure that all passengers can be taken away and to save costs to the greatest extent. These can be calculated by mathematical modeling. Wei Yongsheng said that their mathematical modeling team has a history of six years, and they have cooperated well with each other and have done dozens of projects, large and small. Their entrepreneurial philosophy is to provide unprecedented mathematical modeling optimization and mathematical model solutions for direct customers and potential customers, and truly maximize the investment income and minimize the production cost for customers.

Ten algorithms should be mastered in mathematical modeling

1, Monte Carlo algorithm (this algorithm, also known as random simulation algorithm, is a calculation method to solve problems through computer simulation, and at the same time, it can test the correctness of its model through simulation, which is a necessary method in the competition) 2. Data fitting, parameter estimation, interpolation and other data processing algorithms (there are usually a lot of data to be processed in the competition, and the key to processing data lies in these algorithms. Usually use Matlab as a tool) 3. Linear programming, integer programming, multivariate programming, quadratic programming and other planning problems (most of the problems in modeling competitions belong to optimization problems, and in many cases these problems can be described by mathematical programming algorithms, usually realized by Lindo and Lingo software) 4. Graph theory algorithms (these algorithms can be divided into many kinds, including shortest path, network flow, bipartite graph and other algorithms, and problems involving graph theory can be solved by these methods, which need careful preparation. 5. Computer algorithms such as dynamic programming, backtracking, divide-and-conquer algorithm, branch and bound (these algorithms are commonly used in algorithm design and can be used in competitions in many occasions) 6. Three non-classical algorithms of optimization theory: simulated annealing method, neural network and genetic algorithm (these problems are algorithms used to solve some difficult optimization problems, which are very helpful to some problems, but the algorithms are difficult to implement and need to be used carefully) 7. Grid algorithm and exhaustive method (grid algorithm and exhaustive method are the best algorithms for violent search and have applications in many competition questions. When focusing on the model itself and ignoring the algorithm, this violent scheme can be used, and it is best to use some high-level languages as programming tools. 8. Some continuous discretization methods (many problems are practical, data can be continuous, and computers only recognize discrete data. Therefore, it is very important to discretize it and implement the idea of replacing differential with difference and integral with sum. 9. Numerical analysis algorithm (if high-level language programming is used in the competition, some commonly used algorithms in numerical analysis, such as solving equations, matrix operation, function integration, etc., need to be adjusted by writing additional library functions) 10, image processing algorithm (there is a kind of problem related to graphics in the competition, even if it has nothing to do with graphics, there should be many pictures in the paper. How to display and process these graphics is a problem that needs to be solved, and it is generally used.