1 Mathematical modeling needs abstract thinking.
By analyzing the establishment and solving process of the above model, we can find that abstract thinking and thorough understanding and analysis of the basic concepts of higher mathematics are indispensable in solving problems.
When solving the 1 problem, we closely combined the concepts of "absolute water inflow" and "relative water inflow", analyzed every information contained in the concepts, found the calculation formulas of "absolute water inflow" and "relative water inflow", and established the mathematical model I.
It can be seen that complex practical problems should be summarized into related concepts and definitions of higher mathematics, and calculation formulas should be found by using definitions, so as to establish mathematical models. Abstract thinking plays a key role in this process of layer-by-layer analysis. It is this layer-by-layer analysis that enables complex problems to be solved. So mathematical modeling needs abstract thinking.
2 Mathematical modeling needs to simplify thinking
The so-called simplified thinking is to simplify complex problems and then highlight the essence. Just like X-rays, the flesh and blood are removed, leaving only the skeleton. Only by grasping the main contradiction quickly, abandoning the secondary factors and finding the essence of the problem can we "see through" the essence of the problem.
For example, to distinguish whether a mine belongs to a "low-gas mine" or a "high-gas mine", in essence, we need to make clear the "absolute discharge" and "relative discharge" first, and then compare them with the standard values; The possibility of coal mine explosion is actually a probability problem; The optimal (total) ventilation required by this coal mine is essentially an optimization problem, that is, a linear programming problem with constraints.
This simplified thinking has profound characteristics. It is not innate, but can be formed through careful training and strengthened through hard exercise. In the teaching process of advanced mathematics, it is necessary to cultivate students' profound insight.
3 Mathematical modeling needs critical thinking
After the mathematical model is established and solved, we need to analyze the obtained results, evaluate the established mathematical model, and improve the model in time to obtain the best results. At the same time, we should also point out the practical significance of this model and popularize it. All these links need good critical thinking.
In the teaching process of advanced mathematics, we need to cultivate students' critical thinking. After each problem is solved, we should carry out this kind of reflection training after solving, and keep asking questions: Is the result right? Realistic? What are the advantages and disadvantages of this solution? Is there a better solution? How to improve? Can it be promoted? ..... In this training process, students' critical thinking will be strengthened and improved.