Mathematical modeling: the mathematical activity process of solving problems by establishing mathematical models;
Model thinking: Model thinking is essentially the thinking of looking at the external world from a mathematical point of view and applying mathematics to solve problems in the external world. It emphasizes the connection between mathematics and the outside world.
The mathematical model has two main characteristics: one is the pure mathematical relationship structure formed by abstracting some non-essential attributes of objects; Secondly, this structure is represented by mathematical symbols and can be used for mathematical operations. Equations, inequalities and functions are important mathematical models in junior high school algebra.
The process of mathematical modeling is essentially a "mathematical" process.
The model thought embodies the consciousness and idea of applying mathematics to solve problems. The concept of model is mentioned many times in the standard:
Go through the process of abstraction, operation and modeling of number and algebra (the general goal of number and algebra);
The process of expressing quantitative relations by algebra, equations, inequalities, functions, etc. We can understand the idea of the model; Empirical equation is an effective model to describe the quantitative relationship in the real world (algebraic segment goal);
Combined with the actual situation, experience the process of designing solutions for specific problems and implementing them, and experience the process of establishing models and solving problems. (Content standard of Synthesis and Practice)
From the above introduction, it can be found that from the perspective of mathematical activities, the core step to help students initially form model ideas is the "mathematical modeling" activity. And this activity process can be simplified as the following three links:
1. abstract mathematical problems from real life or specific situations;
(that is, finding and asking questions is the starting point of mathematical modeling)
2. Establish equations, inequalities and functions with mathematical symbols to represent numbers in mathematical problems.
Quantitative relationship and change law;
That is, students should learn mathematics through observation, analysis, abstraction, generalization, selection and judgment.
Move, complete the pattern construction, get the model). This is the most important part of modeling;
3. Solve the model, get the result, and use this result to explain and discuss its significance in practical problems.
Such three links should be embodied in the teaching process of equations (groups), inequalities (groups) and functions.
In the process of helping students form model ideas, we should pay attention to:
The concept of 1. model needs teachers to gradually infiltrate in teaching and guide students to feel constantly.
2. Let students experience the mathematical activity process of "problem situation-establishing model-solving and verifying"
3. Implement "Mathematical Modeling" activities through various learning methods.
It takes a long time for students to really understand the model idea. In this process, students always accumulate experience from relatively simple to relatively complex, from relatively concrete to relatively abstract, master modeling methods, and gradually form the habit of using models for mathematical thinking. In the teaching process, students should be guided to use functions, inequalities (groups), equations "groups", geometric figures, statistical tables and so on to analyze and express practical problems and solve practical problems.
The process of problem situation-modeling-solution and verification can be organically carried out in combination with relevant course content. For example, regarding the teaching of equations, we used to focus on "pure" knowledge and skills from concept to concept, such as equation definition, types, solutions, and discussion of the same solution. Now, we can let students abstract the model of "equation" from various realistic situations, so as to solve specific problems. The process is as follows: