Ways and methods of integrating competition mathematics content into conventional teaching

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A preliminary study on the combination of mathematical modeling competition and conventional teaching

This paper expounds the significance of combining mathematical modeling contest with conventional teaching, constructs the teaching operation system of combining mathematical modeling contest with conventional teaching, and finally puts forward three suggestions on how to overcome the difficulties in teaching.

Keywords: teaching reform, mathematical modeling, mathematics curriculum

First, the significance of the combination of mathematical modeling competition activities and conventional teaching

1 actively adapt to the needs of the times when mathematical technology is widely used.

In the era of increasing mathematicization of science and engineering, the support of mathematical technology is increasingly needed. In fact, with the extensive use of mathematical operation software, computers have become the main operation tools for engineers to apply mathematics to solve engineering problems. Carrying out a series of activities combining mathematical modeling competition with regular teaching can better meet the needs of students for mathematics education, especially from the demand for manual deduction and operation ability training to the demand for modeling, solving and argumentation ability training combined with computer software.

2. Effectively consolidate the teaching needs of the theoretical basis of modeling competition activities.

Mathematical modeling competition requires students to observe and study the internal characteristics and laws of actual objects through investigation and data collection, grasp the main contradictions of problems, establish quantitative relations reflecting actual problems, and then analyze and solve problems by using mathematical theories and methods. It requires students not only to have keen insight, imagination, strong interest in practical problems and extensive knowledge, but also to have a deep and solid mathematical foundation. The value of doing a good job in conventional teaching lies in providing a rich theoretical basis for students to participate in mathematical modeling competitions. In fact, advanced mathematics, linear algebra, probability statistics, mathematical equations, discrete mathematics, graph theory, optimization methods, stochastic processes and other courses, as well as computer application, advanced programming language, object-oriented programming, C language, computer network technology, data structure, JAVE language and other courses, are all conducive to students mastering a lot of mathematics and computer knowledge.

3. The need to promote the deepening and application of conventional teaching reform

The combination of mathematical modeling contest activities and conventional teaching will effectively promote the in-depth development of conventional teaching reform and help to promote the transformation of mathematics teaching methods in science and engineering universities, such as actively introducing software computing into daily mathematics teaching and integrating mathematical modeling ideas and methods into basic courses such as calculus. From the knowledge point of view, the combination of mathematical modeling contest activities and conventional teaching can promote the deepening and application of conventional teaching, mainly in the following aspects: (1) Examples in mathematical models are close to life, intuitive and interesting, so that students can have a deeper understanding of the connotation of mathematics, realize the importance of mathematical learning and further promote the enthusiasm of learning theory on the basis of understanding these examples; (2) When solving mathematical model problems, the mathematical methods used not only cover the advanced mathematics, linear algebra, probability theory, mathematical statistics and other mathematical contents that students have learned in the past, but also may involve some new mathematical knowledge (such as graph theory and combinatorial mathematics, operation and optimization, etc.). ) to be studied; (3) Various mathematical softwares (such as MATLAB, Mathematica, Maple, etc.). ) what to learn in the process of mathematical model competition. It can be seen that these mathematical softwares are not only tools for applying mathematics in future work, but also good tools for students to study and study mathematics in different stages.

The second is the teaching operation system that combines mathematical modeling competition activities with regular teaching.

After years of exploration and practice, combined with conventional teaching, a four-layer pyramid-shaped mathematical model competition was initially constructed.

1 Infiltrate the idea of "mathematical modeling" into public mathematics courses ―― Grassroots units integrate the idea and method of mathematical modeling into the teaching of public mathematics courses in the first and second grades, which promotes the reform of the teaching content and system of college mathematics courses. First, optimize the teaching content of the course and pay attention to cultivating students' comprehensive ability to apply mathematical knowledge, establish mathematical models and apply computer technology to solve practical problems; The second is to reform teaching methods, pay more attention to practical problems, and deduce relevant mathematical concepts, mathematical conclusions and calculation methods through mathematical modeling, so as to cultivate students' abstract thinking ability and make them understand the application background of mathematical knowledge. Teachers who have taught linear algebra know that it is difficult for students to understand the definition of matrix multiplication for the first time. If the definition of matrix multiplication is introduced through some examples, such as product cost, this teaching difficulty can be solved and students' mathematical understanding can be promoted. Another example is advanced mathematics: the concept of derivative can be introduced through the speed of linear motion; The concept of definite integral is introduced by the area of curved trapezoid or the distance of variable speed linear motion. In probability statistics, the concept of variance can be introduced through the average life of products or the average score of shooting; The concept of expectation can be introduced by the deviation of product life and average life of random variables. Stochastic process: Markov process can be introduced through the stability of cross breeding; In mathematical equations, the wave equation can be derived by studying the tiny lateral vibration of bounded strings. Through these cited examples, students can clearly see that the definitions and algorithms they have learned are not symbolic games, but come from the needs of some practical problems, and realize that all knowledge points in abstract mathematics have practical sources and application backgrounds, and future professional learning can not be separated from these "abstract and difficult mathematical symbols", thus stimulating the enthusiasm and initiative of learning.

2. Set up the practical course of "Mathematical Experiment" independently-the promotion layer.

In order to cultivate students' innovative consciousness and ability to solve practical problems by themselves, a centralized practical teaching course of mathematics-mathematical experiment course was set up for senior one and senior two students, 16 class hours. This course is very different from the traditional mathematics course in thinking method, teaching mode and teaching content. We combine teaching with computer, and through the teaching process of "starting from examples (including examples designed by students themselves), using software (including existing mathematical software) to do a lot of experiments on the computer, looking for possible laws, guessing, proving and demonstrating", we take practical problems as the carrier. Combine mathematical knowledge, mathematical software and computer application organically, emphasize students' dominant position, and under the guidance of teachers, use the learned mathematical knowledge and computer technology to analyze and solve some simplified practical problems with the help of relevant mathematical software, and write experimental reports or papers. Starting with the creation of problem situations, teaching can stimulate students' interest and passion in inquiry, guide students to explore the process of learning and experiencing knowledge independently, return to real scientific thinking activities, encourage questioning and criticism and express independent opinions through teacher-student interaction and two-way communication, and cultivate college students' innovative thinking and ability. Through this research-oriented classroom teaching mode, students can fully feel, understand and master the essential content of "mathematical experiment" and get all-round training in mathematical application, computing ability and creativity. In this process, students' interest in learning is improved, and their learning initiative is fully exerted, thus cultivating their initiative, comprehensive application ability and innovative consciousness, which truly embodies the teaching principle of "teacher-oriented and student-centered". In the process of teaching and learning, students actively participate in learning activities, creating a good thinking situation and atmosphere for solving problems and collaborative learning, which is particularly impressive and helps to understand the essence of mathematical knowledge. 3. Offering the elective course of "Mathematical Modeling" independently-offering 48 hours of elective course of mathematical modeling to sophomores. The main purpose of mathematical modeling course is to let students learn to use mathematical knowledge and computer technology flexibly and explore mathematical models to solve some practical problems. The aim is to improve the basic quality and innovative practical ability of college students, and to improve their comprehensive ability to solve practical problems by mathematical means. Traditional college mathematics courses, such as advanced algebra, calculation methods, statistical methods, optimization methods, etc., are mainly taught by teachers in related fields; Mathematical modeling course is not taught by teachers in theory, nor is it computer technology or algorithm, but through teaching, it inspires and guides students to learn how to use mathematics, master how to apply elementary mathematics, advanced mathematics, probability theory, mathematical statistics, analytic hierarchy process and other knowledge to a practical problem in production and life, and combine with other professional theoretical knowledge to complete data collection, conditional hypothesis, scheme demonstration, model design and programming calculation, and can write and solve problems. One is to use, the other is to learn, and their goals are different. Mathematical modeling course covers a wide range and is practical. Cultivating students' innovative spirit and imagination in the whole teaching process is very beneficial to improve students' ability to take part in work and solve practical problems in the future.

4. Organize selection and training to participate in the national mathematical model competition for college students-improve the level.

Select contestants from the students who have participated in the elective course of mathematical modeling and organize pre-competition training. With the deepening and popularization of mathematical modeling activities, the number of students participating in the national mathematical modeling competition for college students in our school has also increased year by year in recent years (see figure 1). In the extracurricular innovation activities of mathematical modeling, we mainly focus on the cultivation of students' autonomous learning, research, innovation, organization, cooperation and expression ability, and require three students to complete a paper together. The topic can be given by the teacher or selected by yourself from actual production and life. After consulting literature, discussing with each other, inspiring each other, arguing with each other, coordinating with each other, working together, seeking common ground while reserving differences, giving full play to their respective strengths, and completing a paper together, the teacher finally makes comments and summaries (the instructor gives specific suggestions and ideas according to each question made by each group). This way of team fighting cultivates students' subjective initiative, creativity, team spirit and coordination ability, which is very helpful for taking part in work after graduation and is the only way to apply mathematics to practice.

From 1998 to now 1 1 year, our students have won the first prize lo and the second prize 15 in the American (international) mathematical model competition for college students: 27 won the first prize and 27 won the second prize in the national mathematical model competition for college students. 1994 to 2007, the number of teams that won the first prize in the Mathematical Model Competition for College Students in our school ranked 1 in Jiangsu Province and ranked 8th in China (data comes from the website of Mathematics China); Since our school participated in the American (international) Mathematical Model Competition for College Students in 2004, the competition results have been among the best in Jiangsu universities, and the practice of mathematical modeling has become a beautiful landscape in the extracurricular innovative practice activities of school students.

Under the mutual promotion of the mathematical modeling competition and the teaching reform of college mathematics course, a practical teaching mode of students' independent exploration and application of mathematical knowledge is constructed through the centralized practical teaching of mathematics and the development of extracurricular mathematical innovation activities. According to the four teaching stages of foundation, popularization, deepening and improvement, all-round and multi-level mathematical practice and innovation activities are carried out to truly improve students' comprehensive quality and innovation ability.

Three. Problems and suggestions

There are many advantages in combining mathematical model contest with conventional teaching, but under the current teaching system, there are still some problems worthy of attention, such as: there are not many teaching units for teachers to combine courses and professional expertise, it is difficult to add new content in class hours, and the problems faced by teachers are more realistic. Take our school as an example. On the one hand, the number of students participating in competitions has increased rapidly in recent years. For example, in 2007, the number of participants was more than five times that of 2000. However, in the past 10 years, the number of instructors has always been only 5-7, and even fewer people have been engaged in this work for a long time. They all have fixed teaching jobs, so the extra time available for mathematical modeling is limited. In addition, the mathematical modeling competition itself needs a group of teachers with solid mathematical foundation and strong interest in mathematical modeling, and it is best to have some practical experience in mathematical modeling and good foreign languages. In particular, they should be enterprising and dedicated to form a coaching team. However, the reality is that quite a few teachers of main courses don't know much about mathematical modeling. To solve the above problems, we need to care about and support the multidisciplinary efforts in mathematical modeling competition and conventional mathematics teaching, face the problems directly, think positively and be brave in practice. Based on 10 years' practice of combining mathematical modeling contest activities with regular teaching, the author's suggestions on solving the above-mentioned related problems and overcoming related difficulties are as follows: 1 Overcome the limitation of class hours, improve classroom teaching methods, strengthen the optimized application of modern teaching media and technology in the classroom, effectively improve classroom efficiency and benefit, and at the same time strengthen the guidance of students' autonomous learning methods, and pay attention to effectively extending some contents in the classroom to extracurricular activities. For example, although mathematical equations have many contents and limited class hours (32 class hours), due to the improvement of teaching and learning methods and means by teachers and students, students' interest in teaching content has increased, and many students have independently extended their study time after class, and independently studied the application of mathematics in physics, which partially made up for the lack of class hours. The popularization of mathematical models can be carried out in schools or online competitions, and promoted and initiated in the name of schools. It is suggested that "Mathematical and Computer Application Ability Competition" and "Learning Competition" related to mathematical model competition should be set up, which should be included in the work plan of relevant departments of the school, and the competition design and organization should be done well to guide the school to establish a good learning atmosphere. 3. The construction of teaching staff should focus on training young teachers to participate in the teaching and training process of mathematical experiments and mathematical modeling; Pay attention to improving the competition guidance level of instructors, and build a platform for them to participate in the training exchange of national backbone teachers; pay attention to strengthening the consciousness of "combining teachers with students" and "learning from each other" so as to integrate teachers' teaching and scientific research. Especially through the cooperation of teachers' teams, teachers in relevant teaching and research departments (especially teachers in mathematics department) can organize students to continue learning further achievements, which can not only cultivate high quality.

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