Author (Source): Zuoshi Middle School Release Date: September 6, 2007

Three teaching forms of mathematical modeling in senior high school

Zuo Shuangqi * (Yu Wei High School)

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The teaching practice of mathematical modeling has been explored in China for more than ten years, and the content of mathematical modeling has been included in the compulsory content of students in the new national curriculum standards and new textbooks. In the exploration of inquiry learning, some schools choose mathematical modeling as a breakthrough; Mathematical modeling is one of the important forms of learning and teaching practice in mathematics. In recent years, in order to cooperate with Shanghai middle school students' mathematical knowledge application competition, our school has made active explorations in mathematical modeling teaching. In view of the difficulty of artificially separating the teaching of mathematical modeling from the daily classroom teaching, and the fear of teachers and students for this diverse and novel learning form of mathematical modeling, our school mainly adopts the following three different levels of teaching forms to overcome the above difficulties.

Research methods and processes

First, the application of mathematical modeling teaching in regular classroom teaching

Broadly speaking, all mathematical concepts, mathematical theoretical systems, mathematical formulas, equations and algorithm systems can be called mathematical modules. For example, "the equation and image of an ellipse" is a mathematical model, and "finding the approximate solution of the equation by dichotomy" is also a mathematical model. In view of the difficulty that students can't abstract, simplify and assume variables and parameters in mathematical modeling to form a clear mathematical framework, we consciously choose appropriate teaching content in conventional mathematics classroom teaching, imitate the process of establishing mathematical models in practical problems, and deal with the conventional learning content in textbooks, thus laying a foundation for students to establish models from practical problems.

For example, in the teaching of dihedral angle, students have the impression that the dam surface forms an appropriate angle with the horizontal plane in their original life experience; I have the impression that the half-open door forms an angle with the wall, so when we let students form the concept of dihedral angle, we should abandon concrete objects such as dams and doors from the students' existing knowledge and abstract that "the figure composed of two semi-planes starting from a straight line is called dihedral angle". Here, the half plane is an idealized object obtained through reasonable assumptions relative to concrete objects such as dams and gates. When we further study how to measure the dihedral angle, we ask students to put forward various schemes, and then discuss and compare whether the geometric quantity defined by each scheme is unchanged for a given dihedral angle, and at the same time concisely express the closure degree of two half planes in the dihedral angle. The above teaching process about the concept of dihedral angle and its measurement method is actually the process of establishing mathematical model and studying model.

This teaching case shows that in the routine daily classroom teaching, it is entirely possible to choose appropriate content and create a teaching scene of mathematical modeling to deal with the teaching content, thus creating conditions for students to truly face practical problems and establish and study models.

Second, teachers provide mathematical modeling teaching problems

The problem mathematical modeling provided by the teacher is basically the same as the modeling tasks that need to be completed in the current mathematical modeling competition for college students and middle school students. Students in this form of mathematical modeling do not need to choose their own practical problems to study, but teachers choose practical problems suitable for students' level to present to students. Under the guidance of the teacher, the student group completed the process of model selection, establishment, calculation and verification by discussion, and finally presented its research results in the form of small papers. In this form of mathematical modeling, students really come into contact with practical problems and experience the whole process of modeling.

Through the teaching of mathematical modeling in daily classroom teaching, students have a certain understanding of what mathematical modeling is and have experienced the exercise of abstracting a clear mathematical framework from specific problems. Therefore, in this form of mathematical modeling teaching, we mainly strengthen the following aspects of teaching.

The practical problems provided by 1. must be moderate in difficulty and suitable for students' cognitive level. For difficult problems, we often give necessary hints, such as inspiring students to turn complex problems into problems that can be modeled by putting forward reasonable assumptions; By prompting students to set relevant variables, the model can be easily established.

Teachers can control the difficulty from the selected practical problems, model assumptions and variable settings, among which model assumptions and variable settings are the key factors that directly affect the establishment of the model, and teachers' failure to plan appropriate teaching forms for this key point is the key to the modeling teaching of "teachers giving questions".

2. In the practice of mathematical modeling, students will experience the whole process of modeling, and in the process of solving the model, they often need to choose the appropriate mathematical software with the help of computers and solve the model through mathematical experiments. In recent years, our school pays more attention to the teaching of this link. Every year, we will give students who attend the summer camp of mathematical modeling for middle school students in Shanghai a tutorial class using the mathematical software Matlab. By making students master the use of one software skillfully, students are guided to learn the use of several other mathematical softwares by themselves, thus paving the way for students to find the solution of the model correctly.

3. In the process of tutoring students in recent five years, we think we can use the following questions to train students' mathematical modeling ability: (1) road and bridge problem, (2) driving problem in limited area, (3) traffic light management problem, (4) inscribed polyhedron problem of ball, (5) helix problem, (6) shortest path problem and (7).

4. In the practice of mathematical modeling, the students' research achievements must be expressed by papers and their own research ideas and achievements, which is the embodiment of a student's comprehensive quality. Because the writing of mathematical modeling papers has certain format requirements, of course, this format requirement is to better let the author show his research results and also to ensure the quality of the papers. Therefore, we give special guidance to the format of students' papers in teaching. Generally speaking, the format of middle school students' mathematical modeling papers should have the following forms.

(1) Summary: Do what? In what way? With what tools? What is the conclusion? Why use this tool? What is the popularization and application of the obtained results?

Keywords: several words used to reflect the main characteristics of the paper.

(2) Restatement of the problem: Restate the problem in your own language and have your own understanding.

(3) Necessary assumptions or assumptions: (1) Assumptions based on actual conditions should be reasonable and simplify the original problem; (2) Definition and declaration of variables.

(d) Problem analysis: What is the relationship between variables? What is known? What needs to be solved mathematically?

(5) Model: If you can write mathematical expressions, you must write them. Several different models can be used.

(6) Model solution: draw a conclusion through various means, including calculators and computers.

(7) Discussion on the problem: the advantages and disadvantages (accuracy and limitations) of the model and the tools used, and whether the conclusions and methods used can be extended to other fields.

(8) Appendix: cited original materials, written procedures, etc.

Counseling students from the above eight aspects and putting forward requirements will effectively ensure students to correctly express their research results in their papers.

Thirdly, students choose the problem of mathematical modeling teaching.

With the first two forms of modeling teaching. After students have a certain level of modeling, they can enter the teaching stage of mathematical modeling for students' self-selected problems. At this stage, students are required to choose a practical problem according to their own modeling knowledge and experience, solve it by establishing a mathematical model, and finally reflect their research results in the form of a paper. If the teaching practice of mathematical modeling is carried out well at this stage, the courage and rich imagination of students to solve practical problems will be unexpected for our teachers. In recent years, our school has strengthened the guidance in the following three aspects in this practical form of modeling teaching.

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