The first question is, what is mathematics? There are different answers to this question from different angles. Generally speaking, the classic definition is that mathematics is a science that studies the quantitative relationship and spatial position relationship of objective reality. My answer is that mathematics is a language. We know that the function of language is communication. How to communicate? Communicate through description. In other words, the function of language is mainly embodied in "description", and the purpose of communication is achieved by describing the object to be expressed. So what does the language of mathematics describe? In what way is it described? What kind of system is it? Next, I will answer these questions step by step and propose how to cultivate mathematical thinking.
First of all, mathematics is a description and summary of objective reality. What is mathematics? As the name implies, mathematics is the study of numbers. Mathematics describes the objective quantitative relationship and spatial position relationship. Mathematics is a quantitative description of the real world and an abstract summary of objective laws. So the first step is to perceive and understand mathematics in real life. In other words, the first step is to learn mathematics intuitively and directly.
Children generally know the world through intuition, that is, they perceive the world directly and intuitively, rather than analyzing it. For example, if we ask children why they are afraid of fire, they will hold out their burned hands and say they are hot. This is the most direct feeling. Generally speaking, primary school students learn mathematics mainly through direct perception. Think about how we learned mathematics for the first time when we were children. I learned 1+2=3 when I was a child. How should I learn? Take an apple, add two apples and count three apples. So 1+2=3. All this needs to be felt and recognized through real things. I remember we learned arithmetic by counting our fingers when we were young. I learned a lot later, and my fingers were not enough. What should I do? I'm very clever, so whenever I meet an arithmetic problem, I pick up a pile of pebbles and count them.
Understanding the world through intuition is the most primitive instinct. It can even be said to be self-taught. So even people who haven't read books know how to calculate the addition, subtraction, multiplication and division of numbers. Primary school students learn mathematics mainly through this vivid and intuitive way. Now that we are middle school students, the corresponding methods of learning mathematics have also changed.
As we know, mathematics in primary schools mainly deals with specific numbers. In middle school, I got rid of specific numbers and mainly dealt with abstract letters and mathematical symbols. The main body of middle school mathematics is algebra and geometry, and geometry is mainly expressed and operated by letters. Algebra, as its name implies, is to replace numbers, and to replace concrete numbers with abstract letters and symbols. From this point, we can clearly see the essential difference between middle school mathematics and primary school mathematics. Many people think that middle school mathematics only learns more knowledge than primary school mathematics, but it is not. From elementary school mathematics to middle school mathematics, the most important thing is not that knowledge, but the change of thinking mode, from intuitive thinking to abstract logical thinking.
Now calculate 1+2=3. Of course, we don't have to count our fingers any more, and we can get the answer directly. On the one hand, practice makes perfect because of a lot of computing experience. On the other hand, it is because we can calculate directly through abstract logic without intuition and external concrete images. At this time, we have reached the second level, and we should treat mathematics abstractly and logically.