George Polya, an academician of the National Academy of Sciences and a famous mathematician, has been devoted to studying the general laws of mathematical thinking. He wrote a book on how to solve problems, which was very popular with readers. This book is to illustrate the application of mathematical thinking in life.
We need to solve all kinds of problems in life every day, from food, clothing, housing and transportation to life choices. The thinking method provided by Paulia in how to solve problems can help us solve these problems better.
Paulia said in the book:
What we usually call insight, judgment, creativity and thinking ability can actually be improved by constantly imitating and practicing problem-solving skills.
Next, let's talk about Paulia's thinking steps to solve the problem.
This process sounds normal, but it is actually very important. Because the process of understanding the topic is the process of setting goals. The clearer your goal is, the more you know what strategy you should use to deal with this problem next, and at the same time you can pay more attention to the process of solving the problem.
But how can we really understand the topic?
Paulia introduced a very useful method of concentration to help us understand the topic faster.
One is analogy.
For example, Einstein's description of time is an analogy. He thinks that time is like a coordinate axis in space, with length, direction and scale. But is this the true state of time? Not necessarily. However, only through this analogy can we understand the strange thing of time with the familiar thing of space.
Second, with the help of graphic method.
The third is the decomposition and recombination method.
Why do you want to switch observation? Because if you go into the details, you may get lost in the details. They will prevent you from paying enough attention to the main points, or even make you completely turn a blind eye to the main points.
But the difficulty is that we can't know in advance which details are ultimately necessary and which are not. If you don't look at the details at all and only consider the whole, you may not be able to understand the topic in depth.
Therefore, it is wise to switch perspectives. Observe the topic as a whole, and then observe the details. One detail touches you, so you focus on it and then observe another detail. After every detail is observed, you will return to the whole.
Finally, we should combine different details to see if we can get new results.
Below. Let's look at a specific topic, how to apply these methods to it?
The same purpose of the three methods is to visualize the whole theme as clearly and vividly as possible.
In addition, Paulia also reminded us that you should pay close attention to the problem when understanding it. What? Is the unknown in the topic.
The problem is this: a bear walks one kilometer due south, then changes direction, walks one kilometer due east, and then walks one kilometer due north. This time just returned to the starting point. Excuse me, what color is this bear?
This question doesn't sound complicated, but the answer is not easy to come up with. Let's analyze it together. What is the unknown number of this question? This is the color of a bear. But how do you get the color of the bear from the mathematical data? This is a little strange.
Then let's look at the known quantity again and get into the details. You can draw a picture on paper, and you will find that under normal circumstances, the starting point can only be answered by one kilometer south, one kilometer east, one kilometer north and one kilometer west. Why did this bear only walk three kilometers and return to the starting point as described in the title?
This contradiction is the real unknown of this problem.
Then we combine these details, take three lines and go back to the starting point, which should be a triangle.
Southeast and northwest, can you walk out of a triangle?
I see. Standing at the North Pole, it's south in any direction, and north in any position. In this way, the bear can start from the North Pole, go south, east and north, then take a triangle and return to the starting point.
Since it is near the North Pole, this bear should be a polar bear. Of course, it is white.
It can be seen that if we grasp the unknown, we will grasp the key to the problem. Analogy, drawing, decomposition and reorganization can help you better understand the topic.
Ideas are invisible and intangible. Is there a routine available? Paulia has provided us with two very useful tools.
The first tool is "special case"
When you have no ideas, you might as well use special cases to help yourself think.
A general question often gives people a feeling that they can't grasp, can't start, and can't grasp anything inside. This is because there are too many conditions, and it seems that nothing can be started. As the saying goes, the profligacy of flowers is becoming more and more attractive. General problems are often uncertain. This uncertainty will become an obstacle to thinking.
So how to reduce this uncertainty? We can consider a special case first, so that we can determine the conditions of the problem and help us explore the internal structure of the problem.
The second tool is "reverse thinking"
When you feel lost in positive thinking, try to deduce it backwards.
Many people admire the power of reverse thinking. For example, Charles Munger famously said in "Poor Charlie's Collection": "Always put yourself in the other's shoes. 」
Let's look at this problem. You are standing by the river, with two buckets beside you. The big bucket can hold 9 liters of water, and the small bucket can hold 4 liters of water. My question is, what can you do to hold 6 liters of water?
Think about this problem with reverse thinking and push forward from the result, and you will find it easier to get the answer.
How to fill 6 liters of water? You can pour 3 liters of water from a 9-liter bucket; So how do you pour out 3 liters of water? After there is 1 liter of water in a 4 liter bucket, you can pour out 3 liters of water. How can there be 1 liter of water in a 4-liter bucket? Barrel 9 liters, small bucket 4 liters, 9 liters minus 4 liters minus 4 liters, which is exactly 1 liter.
Then turn reverse thinking into positive operation, and the problem will be solved. The first step is to put 9 liters into a big barrel, pour 4 liters into a small barrel, and then pour out the water in the small barrel. Step two, pour the water in the big barrel into the small barrel for 4 liters, and then pour out the water in the small barrel. Step 3: Pour the remaining 1 liter from the vat into the small vat, and then fill the vat with water. Step four, fill the small bucket with the water in the big bucket, leaving 6 liters in the big bucket.
In addition, Paulia also proposed a method, that is, keeping an eye on the unknown.
Keeping an eye on the unknown quantity can help you always remember what your goal is in the process of solving problems, which is conducive to enlightening and getting ideas for solving problems.
And how can your attention catch something related to the topic? It depends on the unknown quantity. The unknown quantity is like a wanted order, which turns your attention into a keen detective, and you can catch the suspect from countless thoughts, otherwise you may not notice even if you shake off the key things.
If you make a workbook, it's easy. Turn to the last page to check the answer. However, if it is a practical question and there is no standard answer, how can it be verified?
You can verify your answer in two ways.
One is "dimension verification"
What is a dimension? It is the standard unit of length, area and weight. Many physical problems or geometric problems, we often find an expression, and then how to quickly test it? Just substitute the units of each item in the expression to see if the two sides are equal.
For example, we all know the cuboid volume formula, V=abc, the volume on the left, and the unit is cubic centimeters; On the right, A, B and C are length, width and height, respectively, in centimeters. When they are multiplied, they are cubic centimeters. Such a contrast, both sides are cubic centimeters, this answer is more reliable.
The other is specialization.
Simply put, you have worked out a formula, so we can verify it with specific values.
For example, the volume formula of a cuboid, we can use a specific case, the length, width and height are all 1 cm, and we know that the volume is 1 cubic cm. Substituting into the formula, the result is 1 cubic centimeter.
So dimension test and specialization can actually be combined. If there is a problem with your solution, you can usually find the error quickly by using these two checking methods.
Why turn around?
So how do you review? Paulia has made a list to help you consider your solution from different aspects and find the connection with the knowledge you have gained in the past.
You can look at the longer parts of the solution and see if you can shorten them;
You can re-examine the topic to see if you can see the whole solution at a glance;
You can improve your solution and see if you can make it more intuitive;
You can re-examine the topic and see what details make you have a key idea;
You can also carefully check your own conclusions and see if they can be applied to other topics.
In the process of reviewing, you may be able to find better new solutions and new interesting facts. Even if you don't, if you get into the habit of reviewing and checking your solutions carefully, you will gain some clear and readily available knowledge and improve your problem-solving ability.
Only through active review can you turn the creative means used in the process of solving problems into methods that can be reused in the future.
What's the difference between methods and means?
This used to be Paulia's way of solving problems. In the face of life problems, I don't know if I can learn from them. I hope I can inspire you.