Because "mathematical modeling" is abstract and difficult for junior high school students to understand, the word "mathematical modeling" does not appear in the whole junior high school mathematics textbook. Only in the introduction of chapter 6 "linear function" in the experimental textbook for the eighth grade of compulsory education (Beijing Normal University Press) is it mentioned that "function is a common model to describe the relationship between variables"; In the introduction of Chapter 7 "Binary linear equations", it is mentioned that "equations (groups) are effective models to describe equivalence relations in the real world"; In the introduction of the second chapter of the first volume of the ninth grade, it is mentioned that "like linear equation and fractional equation, linear quadratic equation is also an effective mathematical model to describe practical problems"; In the introduction of the fifth chapter "inverse proportional function", it is mentioned that "function is a mathematical model to describe the relationship between variables". The word "model" really puzzles students, and "building a mathematical model" is even more confusing for students. In essence, the idea of mathematical modeling has been permeated in the arrangement of new textbooks from grade seven to grade nine, especially in equations and functions. Moreover, mathematical modeling can not only cultivate students' good mathematical outlook and methodology, but also promote students to establish practical horizons and concepts and enhance their ability to solve practical problems. Therefore, the author thinks that teachers should do a good job in the enlightenment education of mathematical modeling in teaching, so that students can understand what mathematical modeling is, rather than avoiding it completely.
What is mathematical modeling? When people face a practical problem, they don't directly look for a solution to the problem based on the real matter itself, but through some necessary and reasonable assumptions and simplification, they appropriately use mathematical language and methods to approximate the actual problem, get a mathematical structure (mathematical model), reveal the meaning of the actual problem through the mathematical structure, and return to reality reasonably. This process is called mathematical modeling.
Mathematical model, broadly speaking, all mathematical concepts, mathematical theoretical systems, mathematical formulas, equations and algorithm systems can be called mathematical models; In a narrow sense, mathematical model refers to the mathematical framework or structure for solving specific problems. For example, the binary linear equation is the mathematical model of the problem of "chickens and rabbits in the same cage", the problem of "one stroke" is the mathematical model of the problem of "seven bridges", and so on. Generally speaking, mathematical model is a narrow understanding, especially in junior high school.
Mathematical modeling, in short, is the process of establishing mathematical model, which is a mathematical thinking method, including the whole process of abstracting and simplifying practical problems, establishing mathematical models, solving mathematical models and verifying the solutions of mathematical models.
There is no fixed model for the steps of mathematical modeling, and different people have different views. Now the steps are generally given: including model preparation, model hypothesis, model establishment, model solution, model analysis, model verification and model application, namely
Obviously, it is impossible for junior high school mathematics modeling to have the above complete process. Starting from the actual situation of junior high school students' age characteristics, acceptance ability and knowledge reserve, the modeling of junior high school mathematics should be understood from a narrow perspective, and the requirements should not be too high. As long as students can use their mathematical knowledge to build a suitable mathematical model and solve it. In teaching, students must be the main body, and some unrealistic modeling teaching can not be separated from students.
So, how to make students understand junior high school mathematical modeling?
If teachers can consciously find some good and suitable starting points in peacetime, it will make it easier for students to understand and accept mathematical modeling. For example, the following practical problems in life:
Example 1 A car travels from Shanghai to Beijing along the Beijing-Shanghai Expressway, with an average speed of 100km/h, which takes * *12.65 h. If the average speed is 1 10km/h, how long will it take?
This problem is likely to be solved by elementary school formula:
The distance from Beijing to Shanghai is:100×12.65 =1265 (km).
Therefore, the driving time when the car returns is:1265 ÷110 =1.5 (h).
At this time, teachers should guide students in time: "Equations and functions are also mathematical models that describe the relationship between quantities. How to introduce some practical problems into specific models is the key to whether the problems can be solved, and the consciousness of establishing mathematical models is particularly important. If the equation is used to solve this practical problem, it is to establish an equation model; If we use the idea of function to solve it, it is to establish a function model. This is the mathematical model. "
Students can learn about mathematical modeling from other related disciplines. Because mathematics is a tool for learning other natural and social sciences, and other disciplines are closely related to mathematics. Therefore, we should pay attention to the echo with other subjects in teaching, which can not only help students deepen their understanding of mathematical modeling, but also cultivate their modeling consciousness. This model consciousness is not only abstract mathematical knowledge, but also has a far-reaching influence on their future exploration of other disciplines by using mathematical modeling. For example, the following question is related to physics:
Practical problems in life can also be solved by establishing geometric models, such as engineering positioning, scrap disposal, arch bridge calculation, belt transmission, repairing broken wheels, runway design calculation and so on. , which involves certain graphic properties, often needs to be solved by establishing geometric models. Choosing some practical problems solved by establishing geometric models can also help students better understand mathematical modeling. For example, the following interesting practical question:
As shown in figure 1, in a football match, a player approaches the goal AB with the ball in a straight line. Where is the best place for him to shoot?
Build a model? This is a geometric positioning problem. According to common sense, the best starting position should be the point with the largest opening angle with AB on the straight line. This is the most likely time to score. The problem is transformed into finding the point p on a straight line to maximize ∠APB. For this reason, a circle passing through point A and point B is tangent to a straight line, and the tangent point P is the demand. When the straight line is perpendicular to the line segment PB, it is easy to know that the closer point P is to the goal, the better the start. It can be seen that the kung fu of "closing the door" should include choosing the best location to shoot.
It can be seen that the key to solving practical problems by mathematical modeling is to abstract practical problems into mathematical problems. We must first extract the mathematical model of practical problems through observation and analysis, and then put the mathematical model into a knowledge system to deal with. This requires students not only to have certain abstract ability, but also to have considerable observation, analysis, synthesis and analogy ability.
Students' acquisition of this ability is not a one-off event, and the consciousness of mathematical modeling needs to run through the whole middle school teaching. Junior high school only caught a glimpse of the whole leopard, which laid some foundation for it. In the future, when students enter high school or university to receive mathematical modeling education again, they will find that the field of mathematical modeling is so extensive and harsh.
Many problems in life can be solved by mathematical modeling. As long as we look hard, we can find practical examples to help students understand mathematical modeling. In order to make students fully understand mathematical modeling problems, teachers should choose appropriate practical problems, create reasonable problem situations, do it themselves, collect and sort it out according to local conditions, and turn it into mathematical modeling problems suitable for students and close to their real life. At the same time, we should pay attention to the openness and development of the problem, and create some reasonable, novel and interesting problem situations as much as possible, which will not only let students understand mathematical modeling, but also stimulate their curiosity and thirst for knowledge in exploring mathematical modeling.