With the progress of mankind, the development of science and technology, and the increasing digitalization of society, the application of mathematical modeling is more and more extensive, and the mathematical content around people is becoming more and more abundant. It is of great significance to emphasize the application of mathematics and cultivate the consciousness of applied mathematics to promote the implementation of quality education. The position of mathematical modeling in mathematical education has been promoted to a new height. Solving mathematical application problems through mathematical modeling can improve students' comprehensive quality. Based on the characteristics of mathematical application problems, this paper analyzes how to use mathematical modeling to solve mathematical application problems, hoping to get help and correction from colleagues.
First, the characteristics of mathematical application problems
We often call it a kind of mathematical problem, which comes from the reality of the objective world, has practical significance or background, and needs to be transformed into mathematical form through mathematical modeling, so as to be solved. Mathematical application problems have the following characteristics:
First, the mathematical application problem itself has practical significance or background. Reality here refers to all aspects of the real world, such as production reality, social reality, life reality and so on. For example, practical problems closely related to textbook knowledge and originated from real life; Application problems related to the intersection of modular subject knowledge networks; Application problems related to the development of modern science and technology, social market economy, environmental protection and realistic politics.
Secondly, the solution of mathematical application problems needs the method of mathematical modeling, which makes the problem mathematical, that is, the problem is transformed into mathematical form to express and then solved.
Third, there are many knowledge points involved in mathematical application problems. It is a test of the ability to comprehensively apply mathematical knowledge and methods to solve practical problems. It examines students' comprehensive ability and generally involves more than three knowledge points. If you don't master a certain knowledge point, it is difficult to answer the question correctly.
Fourthly, there is no fixed pattern or category for the proposition of mathematical application problems. It is often a novel practical background, which makes it difficult to train the problem model and solve the changeable practical problems with "sea tactics" Solving problems must rely on real ability, and the examination of comprehensive ability is more real and effective. Therefore, it has broad development space and potential.
Second, how to model mathematical application problems
Establishing mathematical model is the key to solving mathematical application problems. How to build a mathematical model can be divided into the following levels:
The first level: direct modeling.
According to the subject conditions, the ready-made mathematical formulas, theorems and other mathematical models are applied, and the explanatory diagram is as follows:
Conditional translation of themes
In mathematical expression,
Substitute the problem setting conditions of the application test into the mathematical model to solve.
Select the one that can be used directly.
mathematical model
The second level: direct modeling. You can use the existing mathematical model, but you must summarize this mathematical model, analyze the application problems, and then determine the specific mathematical model needed to solve the problems or the mathematical quantities needed in the mathematical model before you can use the existing mathematical model.
The third level: multiple modeling. Only by refining and dealing with complex relations, ignoring secondary factors and establishing several mathematical models can we solve the problem.
The fourth level: hypothesis modeling. Before the mathematical model is established, it needs to be analyzed, processed and assumed. For example, when we study the traffic flow at intersections, we can only model them when the traffic flow is stable and there are no emergencies.
Third, the ability to build mathematical models.
It is the key of the whole mathematics teaching process to establish a mathematical model from practical problems and solve practical problems by solving mathematical problems. Mathematical modeling ability is directly related to the quality of solving mathematical application problems, and also reflects a student's comprehensive ability.
3. 1 Improve analytical comprehension and reading ability.
Reading comprehension ability is the premise of mathematical modeling. Generally, mathematical application problems will create a new background, and some special terms will be used for the problem itself, and immediate definitions will be given. For example, 1999 college entrance examination question 22 gives the process description of cold-rolled steel strip, gives the special term "thinning rate" and gives a direct definition. Whether it can be understood deeply reflects its comprehensive quality, and this understanding ability directly affects the quality of mathematical modeling.
3.2 Strengthen the ability to transform written language narration into mathematical symbol language.
It is the basic work to translate all the words and images in mathematical application problems into mathematical symbolic language, that is, numbers, formulas, equations, inequalities, functions, etc.
For example, the original cost of a product is one yuan. In the next few years, it is planned to reduce the cost by p% on average every year compared with the previous year. What is the cost in five years?
The cost of translating the words given in the question into symbolic language is y=a( 1-p%)5.
3.3 Enhance the ability to choose mathematical models.
Choosing mathematical model is the embodiment of mathematical ability. There are many ways to establish mathematical models, and how to choose the best model to reflect the strength of mathematical ability. The establishment of mathematical models mainly involves equations, functions, inequalities, general term formulas of series, summation formulas, curve equations and other types. Combined with the teaching content, taking function modeling as an example, the following lists the mathematical models selected for practical problems:
Practical problems of function modeling types
Functions of cost, profit, sales revenue, etc.
Quadratic function optimization, material saving, minimum cost, maximum profit, etc.
Power function, exponential function, logarithmic function, cell division, biological reproduction, etc.
Trigonometric function measurement, alternating current, mechanical problems, etc.
3.4 Strengthen the ability of mathematical operation.
Mathematical application problems are generally complicated and have approximate calculation. Although some ideas are correct and the modeling is reasonable, it lacks computing power and will give up all efforts. Therefore, strengthening the reasoning ability of mathematical operations is the key to the correct solution of mathematical modeling. It is not advisable to ignore the cultivation of computing ability, especially computing ability, and only pay attention to the reasoning process without paying attention to the computing process.
Using mathematical modeling to solve mathematical application problems is very conducive to thinking from multiple angles, levels and sides, cultivating students' divergent thinking ability, and is an effective way to improve students' quality and implement quality education. At the same time, the application of mathematical modeling is also a kind of scientific practice, which is conducive to the cultivation of practical ability and a necessary condition for the implementation of quality education, and needs enough attention from educators.
Strengthening the teaching of mathematical modeling in senior high school and cultivating students' innovative ability
Based on the teaching of new mathematics textbooks in senior high school, combined with the compiling characteristics of new textbooks and the development of research-based learning in senior high school, this paper explores how to strengthen mathematics modeling teaching in senior high school and cultivate students' innovative ability.
Keywords: innovative ability; Mathematical modeling; Research study.
"Full-time senior high school mathematics syllabus (Trial)" puts forward new teaching requirements for students, requiring them to:
(1) Learn to ask questions and clarify the direction of inquiry;
(2) Experiencing the process of mathematical activities;
(3) Cultivate innovative spirit and application ability.
Among them, innovative consciousness and practical ability are one of the most prominent features in the new syllabus. Mathematics learning should not only cultivate and improve the basic knowledge of mathematics, basic skills and thinking ability, calculation ability and spatial imagination ability, but also cultivate and improve the ability of applying mathematics to analyze and solve practical problems. It is not enough to cultivate students' ability to analyze and solve practical problems only by classroom teaching. It is an important purpose and basic principle of mathematics teaching to have practice and cultivate students' innovative consciousness and practical ability. In order to make students learn to ask questions, make clear the direction of inquiry, communicate with existing knowledge and abstract practical problems into mathematical problems, it is necessary to establish mathematical models, thus forming a relatively complete mathematical knowledge structure.
Mathematical model is a bridge between mathematical knowledge and mathematical application. Studying and studying mathematical models can help students explore the application of mathematics, arouse their interest in mathematics learning, cultivate their innovative consciousness and practical ability, and strengthen the teaching of mathematical modeling, which is of far-reaching significance to the intellectual development of students. This paper talks about some experiences on how to strengthen the teaching of mathematical modeling in senior high school.
First, we should pay attention to the problem teaching before each chapter, so that students can understand the practical significance of establishing mathematical models.
Each chapter of the textbook is introduced by a related practical problem, which can directly tell students that after learning the teaching contents and methods of this chapter, this practical problem can be solved by mathematical model, so that students will have innovative consciousness, desire for new mathematical model and practical consciousness, and try it in practice after learning.
For example, in the new textbook Trigonometric Function, it is proposed that there is a semi-circular open space centered on point O, and an inscribed rectangle ABCD should be drawn on this open space to turn it into a green book, so that the edge AD of the book falls on the diameter of the semi-circle, and the other two points BC fall on the circumference of the semi-circle. Given that the radius of a semicircle is a, how to choose the positions of point A and point D symmetrical about point O to maximize the rectangular area?
This is a good opportunity to cultivate innovative consciousness and practical ability. We should pay attention to guidance, make an abstract analysis of the practical problems investigated, establish corresponding mathematical models, and put forward new knowledge through old and new ways of thinking, so as to stimulate students' desire for knowledge, such as not dampening students' enthusiasm and losing "bright spots".
In this way, through the problem teaching before the chapter, students can understand that mathematics is learning, researching and applying mathematical models, and at the same time cultivate their awareness of pursuing new methods and participating in practice. Therefore, we should attach importance to the teaching of the problems in the previous chapter, supplement some examples according to the needs of the construction and development of market economy and the problems found in students' practical activities, strengthen the teaching in this respect, make students pay attention to mathematics in their daily life and study, and cultivate their awareness of mathematical modeling.