Mathematics is a game.
What exactly is mathematics for? Why should we study math? Is mathematics really closely related to real life? What exactly do we study mathematics for?
Everyone says that the math courses in China are much more difficult than those abroad. China's basic education is very solid, and China students' mathematics can crush foreign students. The conclusion of my reflection is that the problem in our math class is not difficult, but that there is no math in our math class at all.
Why can foreign schools tolerate what we think is particularly low-level operational incompetence, but they can still train so many excellent mathematicians? Do we study mathematics as an operation skill or as a mathematical thinking? The conclusion of my reflection is that mathematics is a game.
Because mathematics can be divorced from real life and perform logical operations in a completely fictional world, the greatest pride of mathematics is that it can be completely unrelated to real life. Now we have skillfully memorized all the mathematical formulas and geometric principles, and then we continue to use them to do all kinds of problems.
This is equivalent to when we play games, I won't let you experience the process of the game. I'm just telling you, press forward, press five times, then jump, go left, then hit, and finally reach the checkpoint. You remember everything, Dad. You remember everything, and then you can finish the task again and again.
Do you think this game is still interesting? Is this still a game? The purpose of the game is not the result. All the formulas we see are actually the result of the game. No matter how clever it is, it will not help us understand the essence of the game.
What we need is independent thinking and creative thinking, and the last thing we need is "being trained". As Gauss said, we need ideas, not symbols.
The way to learn mathematics is just like watching a movie or reading a book. For example, when we watch Journey to the West, I won't tell you why Tang Priest learned the scriptures and what he experienced. I'll tell you, the Tang Priest finally got the scriptures back after 81 difficulties. You should remember what these 8 1 difficulties are, and this order can't be chaotic. At the end of the exam, you will pass by answering the name of each difficulty.
Do you think this interpretation is correct? If people think it is wrong to read books like this, why is it right to study mathematics like this? In fact, mathematics, like all literature, art and film and television works, is created by human beings for entertainment.
And it is more free than all our games, such as football, basketball and all the movies we watch. Because these games are very dependent on the physical properties of our real world, and mathematics can completely create a world, and the rules of that world can be completely decided by you.
You just need to bring in the conditions of the question you want to answer and use a rigorous reasoning to find the answer. This is a freer and more creative game.
02
Mathematical training is a kind of mathematical thinking.
The biggest problem in our mathematics is that each of us "vaguely" remembers some formulas and definitions, but clearly remembers our "hatred" for them! I said that mathematics is actually a game. You can make rules in your imaginary world to test your creativity.
Let's come up with a problem of "chickens and rabbits in the same cage" I believe all the children have tested this question: how many chickens and rabbits are there in the cage? There are 25 heads and 70 feet in the cage. How many rabbits and chickens are there?
I don't know how you did the problem, but I remember that when I learned this problem, it was the first time I learned a linear equation with one variable. The teacher told me that the standard answer is-let's assume that the number of rabbits is x:
4X+2×(25-X)=70
X= 10
Rabbit = 10 (only)
Chicken = 25- 10 = 15 (only)
Is this the right thing to do It must be right. Is it fun to do this problem? It's definitely not fun. In fact, there are many proofs of "chickens and rabbits in the same cage". Let me give you a particularly interesting example:
Imagine a world. The rabbit and chicken in this cage are in this imaginary world. You stand in front of this cage. At the command, all animals raised one foot. How many animals are standing at this time? 70-25 = 45, and 45 feet to stand.
When you give the order again, all the animals stand with their feet up. At this time, there are still 45-25 = 20 feet standing, but the chicken is already sitting on the ground, because the chicken has only two feet that are all raised and the 20 feet are all rabbits, so the rabbit has 10, and the chicken has 15.
It is interesting to solve this problem in this way. There are many ways to solve this problem:
Suppose a chicken also has four feet, because we don't count its wings as feet. Let's assume that all chickens also have four feet. 25 kinds of animals should be 65,438+000 feet, so why only 70 feet?
Because 30 (100-70 = 30) wings don't count as feet! Those 30 wings must be chickens, so they are 15 chickens.
Let's take another example: there is a triangle in the rectangle. What is the ratio of the area of this green triangle to the area of the blue triangle next to it? Usually, our standard answer is:
Area of triangle = 1/2× base× height.
Rectangular area = base × height
In this way, the area ratio of the triangle inside and outside the rectangle is 1 1. Is this the right solution? Absolutely right! Is this an interesting solution? This is definitely not funny. Let's talk about another solution, or put it in our imaginary world:
Think of it as a piece of paper. We draw a line from the vertex of the triangle and then tear it apart. It becomes two squares, and each square is diagonally divided into two identical parts, that is, the green and blue parts in each square have the same area, and their combined area is also the same, which is 1: 1.
? What is mathematical thinking?
Thinking is an advanced form of psychological activity. Mathematical thinking is what people usually call mathematical thinking ability, that is, the ability to think and solve problems from a mathematical perspective.
Mathematical thinking gives you a pair of eyes to see the world again, and another kind of eyes to see the world. Sometimes it may be quite different from our intuitive understanding of the world. For example, what do you think are the numbers of natural numbers, even numbers and odd numbers?
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1 1, 12 ...
2, 4, 6, 8, 10, 12, 14, 16, 18, 20 ...
1, 3, 5, 7, 9, 1 1, 13, 15, 17, 19 ...
We intuitively feel that even numbers and odd numbers are combined into natural numbers, and natural numbers are naturally more than them. But from a mathematical point of view, their number is the same, because their number is infinite.
Let me give you another example: lose weight. From a mathematical point of view, it is impossible to lose weight. Because if you want to lose 10 kg, you have to lose 5 kg first. If you want to lose 5 kg, you have to lose 2.5 kg first. If you want to lose 2.5 kg, you must lose 1.25 kg first. This is an infinite fold, but people can't lose infinitely much, so it is impossible to lose weight.
It's like a race between a tortoise and a hare. As long as the tortoise runs first, the rabbit will never catch up. Why? Because the rabbit will always get to that point before the tortoise, as long as the rabbit gets to that point, the tortoise will go forward again, so the rabbit will never catch up with the tortoise.
We know this is impossible in real life, but when we think about it, these famous "paradoxes" are just games that mathematicians are playing.
How should I exercise my mathematical thinking? I think every number should be regarded as an element in the game. Forget all the formulas and theorems, prove the problem in the way you want and like, and then keep trying and making mistakes.