First, the understanding of the construction of mathematical models
Mathematics teaching is the establishment of mathematical knowledge model and the application of its methods on a certain basis. Mathematical modeling is an extremely important mathematical thinking method. It has an extremely important influence on students' learning and dealing with mathematical problems. It can help students understand the function of mathematics and arouse their interest in mathematics learning. Therefore, in mathematics teaching, constructing and mastering mathematical modeling methods is an important way to cultivate students' ability.
Mathematical model is an important bridge between the basic knowledge of mathematics and the knowledge of applied mathematics, and it is also an important way for teachers to cultivate students' mathematical ideas and methods in normal mathematics teaching. What's more important is to show the problem through real scenes and create a problem-solving environment to build a model, so as to help students activate their knowledge in the process of solving problems and turn factual knowledge into a problem-solving tool. In the process of exploring and obtaining mathematical models, students also get ideas and methods to build mathematical models and solve practical problems, which is far more significant for students' development than just getting some mathematical knowledge.
The so-called mathematical model refers to the mathematical structure that simplifies and refines mathematical knowledge, and then summarizes and describes it through mathematical language, symbols or graphics, reflecting the relationship between specific problems or specific things. The generalized mathematical model includes various concepts, formulas and theories in mathematics. Because they are all abstracted from the prototype of the real world, in this sense, the whole mathematics can also be said to be a science about mathematical models. In a narrow sense, mathematical model only refers to the mathematical relationship structure that reflects a specific problem or a specific thing system. In this sense, it can also be understood as a mathematical expression of the internal relationship between variables in a system.
Establishing mathematical model is an important task in mathematics learning. Mathematics curriculum standard arranges four learning fields: number and algebra, space and graphics, statistics and probability, practice and comprehensive application, which emphasizes students' mathematical activities and develops students' sense of number, symbol, space, application and reasoning ability. The most important part of these contents is the mathematical model. In primary school, mathematical models are expressed by a series of probability systems, algorithm systems, relationships, laws and axiomatic systems. It can be said that the process of students learning knowledge is actually the process of understanding and mastering a series of mathematical models.
Second, the basic principles of mathematical model construction
1, simplification principle-the prototype in the real world is a multi-factor, multi-variable and multi-level complex system. To some extent, simplifying the prototype is to grasp the main contradiction. The mathematical model is simpler than the prototype, and the mathematical model itself is "the simplest".
2. The principle of derivability-through the study of mathematical model, some definite results can be obtained. If the established mathematical model is mathematically irreversible and cannot be applied to the prototype, then this mathematical model is meaningless.
3. Reflective principle-Mathematical model is actually a reflection form of human beings to the real world, so there should be some "similarity" between the mathematical model and the prototype of the real world. Mastering mathematical expressions or mathematical theories similar to the prototype is the key skill to establish mathematical models.
Third, the method of mathematical model construction
1, the establishment of mathematical model should let students guess boldly, and then make concrete analysis under intuitive circumstances.
Conjecture is an advanced way of thinking with certain intuition, and it is a very important way of thinking for exploratory or discovery learning. Before teaching students some mathematical theorems, we might as well let them guess this theorem boldly according to their existing knowledge. For example, after students have mastered the derivation process and calculation method of the area calculation of rectangular, square, parallelogram, triangle and other plane figures, when teaching the area calculation of trapezoid, I let students boldly guess who its area calculation may be related to. According to the knowledge learned in the past, let the students think of the transformed mathematical thought and speculate that it may be related to the area calculation of parallelogram, and then let the students learn from various trapezoidal materials provided by me and proceed from intuitive graphics.
2. The construction of mathematical model should enable students to make an effective comprehensive comparison among many intuitive or close-to-life examples.
Synthesis means that students sort out and combine the analysis of mathematical phenomena and examples in the process of learning, thus forming an overall understanding of this kind of mathematical knowledge. Comparison is to distinguish the similarities and differences between related mathematical phenomena and mathematical examples. There are many aspects of comparison in mathematics, including the comparison of quantity and size, the comparison of similarity and difference, the comparison of structure and relationship, the comparison of law and nature, etc. The purpose of comparison is to understand the connections and differences between things, to clarify the identity and similarity between things, and to explain the * * * identity model behind things. For example, in the teaching of "percentage in life", I first deduced it from the salt content of the Dead Sea. After giving many related examples, such as attendance, qualified rate, survival rate, qualified rate, germination rate, flour yield and so on. Students come to the conclusion that these are percentages in life and are calculated as percentages of the total. By comparison, although they are all percentages, they are also different. Salt content refers to the percentage of salt in brine, and attendance refers to the percentage of actual attendance to total attendance.
3. To build a mathematical model, students should abstract the characteristics of * * * from concrete examples, and then summarize them with mathematical language or symbols.
Abstraction is to discover the essential characteristics of * * * from numerous mathematical examples or phenomena. And generalization is to summarize and summarize the abstract similarities in mathematical languages or symbols. For example, in the relationship between teaching scores and division, through a large number of examples, students can abstract that their own * * * is: dividend ÷ divisor = dividend/divisor, and finally sum up a÷b=a/b(b≠0) with mathematical symbols.
4. The mathematical model must be fully verified by students, and then it can be effectively applied after reaching a conclusion.
When drawing a preliminary conclusion, students should be given enough space to fully verify it. In the process of verification, we may find new phenomena, and in the process of solving new problems, we will further improve our guesses, and finally find the law and draw conclusions. And use this law to solve more practical problems. This is not only a process of active learning, but also a process of discovery learning and innovative learning. For example, when I was teaching the area of triangles, students made a parallelogram by two identical acute triangles. Through analysis, abstraction and generalization, I proposed that right triangles or obtuse triangles are the same. Students will be fully verified by operation, as long as two identical triangles can be combined into a parallelogram, they all have the above laws. At the same time, students will find that two right-angled triangles are not only parallelograms, but also rectangles, and two isosceles right-angled triangles are not only rectangles, but also special rectangles, that is, squares.
5. Mathematical activities should be the main form of constructing mathematical models.
Because mathematical thinking method is different from mathematical knowledge points, it cannot be replaced by a definition or concept. It has its activity form and rich connotation. Therefore, we should teach mathematical thinking methods in various forms of mathematical activities.
(1) The real life scene of the problem-select the appropriate environmental background and related materials to trigger the discussion.
(2) Reasonable explanation of the problem-choose an appropriate mathematical form and restate it.
(3) The complete solution of the problem-the psychological activity process of the formation of mathematical thinking method, which is mainly solved by cognitive objects or problems.
(4) Mathematical model of the problem-forming cognitive and thinking models, making mathematical concepts or models free from concrete materials, thus promoting the formation of students' mathematical concepts (consciousness).
6. The construction of mathematical model should integrate multiple ways of thinking.
Demonstration-generalization method, similar comparison-abstraction method, intuitive thinking, image thinking, abstract thinking, logical thinking, etc. It should constantly appear in mathematics teaching, so that the teaching process can go through the process of visualization-quasi-modeling-modeling.
The idea of mathematical modeling is different from ordinary mathematical knowledge teaching. It should be: concrete real life scenes-analysis-abstraction-mathematical description-establishing models-forming thinking methods-solving problems (or cognitive formation)-forming concepts (consciousness)-solving more practical problems.
Four, the basic steps of mathematical model construction
The most important thing to solve the problem with mathematical model method is to establish a mathematical model suitable for the problem. There are several basic steps:
1, ask questions and express them in accurate language;
2. Analyze various factors and make theoretical assumptions;
3. Establish a mathematical model;
4. Mathematical deduction is carried out according to the mathematical model, and meaningful mathematical results are obtained;
5. Analyze mathematical conclusions. If it meets the requirements, the mathematical model can be generalized and systematized to solve the problem. If it does not meet the requirements, further discuss, modify the assumptions and rebuild the model until it meets the requirements;
6. optimization. The hypothesis and mathematical model of a problem are constantly modified and optimized. Because there may be several models for a problem or a class of problems, we should compare them until we find the best model.
Mathematical model is a bridge between the basic knowledge of mathematics and its application. The process of establishing and dealing with mathematical models is the process of applying mathematical theoretical knowledge to practical problems. Moreover, it is more important to build a model. Students can realize the excellent opportunity to develop and recreate mathematics from the actual situation. Students can better understand the natural relationship between mathematics and nature and society in the process of establishing models and forming new mathematical knowledge. Therefore, in primary school mathematics teaching, it should be our knowledge to let students learn, do and use mathematics from realistic problem situations. Only in this way can "problem solving" in mathematics teaching have a corresponding environment and atmosphere.