For example, numbers are completely non – existent in the real world and are purely products of human thinking. Are there 1, 2, 3 in the world? In the world, there is only one person, two cars, and three dogs. For a problem like 1 + 2+3, everyone knows the answer. But if you add people, cars, and dogs together, you don’t know what the result means. When humans apply the mathematical logic of 1 + 2 + 3 = 6 to practical matters, it is arithmetic. When humans only reason about the relationships between numbers and symbols, it is mathematics.
Mathematics originated from the practical use of arithmetic. A large number of arithmetic practices inspired ancient philosophers and mathematicians. They set aside all practical functions and reasoned about the relationships between numbers and geometric points, lines, and planes purely at the logical level. The results of these reasonings were then returned to practice to guide the work of craftsmen. It can be imagined that the number of people with this kind of thinking tendency must be very small. Otherwise, if everyone stopped working in daily life and spent all day deducing these things, could humanity still exist? For example, in ancient Greek times, among a large number of wealthy and idle nobles, 90% were playboys who indulged in pleasure – seeking, and only a very few were addicted to such “games”.
Applying the results of mathematical inferences to practice was not their goal, and most craftsmen could neither understand nor read their results. Therefore, the technology in ancient Western countries lagged far behind that of China. In contrast, the Chinese people hardly cared about such things. They only resorted to logical reasoning when they encountered difficulties in practice. So, as long as they could solve the immediate problems, they would stop this kind of logical reasoning. The most famous example is the problem of pi. Western mathematicians quickly knew that pi is the ratio of the circumference to the diameter. But we in China needed it for making wheels.
So, we calculated it accurately to six or seven decimal places. Western mathematicians, who didn’t care about practical applications, stopped researching after knowing it was a ratio. Therefore, their accuracy was far inferior to that of Zu Chongzhi. Zu Chongzhi conducted research with a specific problem in mind. So, after solving the problem, he stopped further reasoning. Thus, we had such an accurate calculation, but we didn’t derive other geometric theorems and formulas. Obviously, “mathematical thinking” must not become the main way of thinking because the majority of people must focus on survival.
Interest and Incentives, Unlocking My “Mathematical Thinking”
This topic is profound and interesting. I grew up in the countryside in the 1960s and 1970s. There were no kindergartens in rural areas at that time, and the educational conditions for compulsory education were very poor. There was no talk of developing children’s intelligence, and the idea of cultivating “mathematical thinking” was simply nonsense. When I was about seven years old, my second brother took me to register at the production brigade. There was no preschool transition at all. I directly entered the first grade, starting to learn Chinese by learning characters and starting to learn mathematics from 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. As a rural child, I was dull and introverted, and my learning ability was too slow. After one semester, I still hadn’t mastered addition and subtraction within ten, and my grades were among the last few. The teacher was at a loss what to do with me.
Throughout the five – year primary school, my math scores were always in the twenties or thirties. In the town – wide primary school graduation unified examination, I only got 13 points in math. The teacher at that time said that I was not a material for school. When I entered junior high school, the math teacher changed. Mr. Li taught us math. At that time, junior high school mathematics was divided into geometry and algebra volumes. The first volume was algebra. Mr. Li’s vivid teaching opened up my dull mind. When Mr. Li said that Q could represent a certain number, it immediately aroused my great interest. It turned out that letters could replace numbers. From then on, I fell in love with math.
Although my handwriting was not good, my homework became neat and earnest, and my math grades got better and better. Unconsciously, I learned the addition, subtraction, multiplication, and division from primary school. In the first math test, I actually got 89 points (out of 100). Mr. Li praised me, saying that I got 13 points in the primary school graduation exam and now 89 points. This was the result of learning with passion and effort.
In fact, I don’t know how this transformation happened. But one thing is certain: interest is the most important. Interest is the best medicine and the best teacher. There is no fixed pattern for developing mathematical thinking. Life is full of different situations, and everyone grows up differently, so the methods are different. The key is whether you encounter an incentive that can arouse your interest in mathematical thinking! My incentive was that letters could replace numbers.
Mathematics is everywhere in life.
When children are young, let them count the number of stairs, the number of fruits, the shapes and sizes of things, etc. When going out, what time do you start? What time do you come back? How many hours in total? When shopping, how many things do you buy? How much is one thing? How much change should you get? The TV tower is very high, and it’s impossible to climb up. How can you measure its height on the ground? If the child doesn’t know, you can tell him that after learning junior high school math knowledge, he can do it. This can stimulate his interest in mathematics. First, let the child understand the taxi – charging method, and then when taking a taxi, let him calculate approximately how much the fare is.
Discounts in shopping malls are also a mathematical phenomenon. For example, if a piece of clothing costs 500 yuan after an 80% discount, what is the original price? Let the child calculate it. The great mathematician Hua Luogeng said, “The immensity of the universe, the minuteness of particles, the speed of rockets, the ingenuity of chemical engineering, the changes of the earth, the complexity of daily life, all use mathematics everywhere.” Mathematics originates from life and serves life. Mathematics is inseparable from life. Parents should always have an educational awareness when accompanying their children, so that children can have a basic understanding of the application of characters and mathematics in life from an early age.
Regular training will turn into mathematical thinking, which is a good foundation for children’s future math learning in school. Of course, these are only supplementary and can play a role in stimulating interest. But ultimately, it still depends on doing more relevant training when learning mathematics to develop a rigorous and high – level mathematical thinking.
Mathematics is different from subjects such as Chinese and history. Learning by “memorization” has very poor results.
Because in math exams, formula memorization is never tested. Only the application and proof of formulas are tested. To learn math effectively, the “understanding” method should be used.
I. Most memorization methods are weak connections
When there are many knowledge points and it’s hard to remember, we often use the “memorization” method to learn. That is, we string together fragmented knowledge points. We can recite ancient poems relying on the connection of sound rhythms. This kind of rhythmic connection is not inevitable, not very clear – cut, but it seems to be quite effective. Similar memorization methods include: image memorization, story memorization, homophone memorization… However, the connections between knowledge points formed by these methods are weak connections and are easily broken. Using these methods in mathematics may be harmful.
Mathematical knowledge mostly consists of strong connections. Many geometric theorems can be proved from each other. From axioms to theorems, it is an inevitable derivation. These derivation processes are accurate and cannot be questioned. This kind of connection is called logical reasoning. These derivation processes are what you should practice, not the conclusions.
II. Because it’s too abstract, it’s hard to understand
Mathematics is abstract. Even for a very concrete geometric shape like a circle, you can’t find an absolute “circle” in the real world. An absolute “circle” only exists in people’s minds. The higher the grade, the more abstract and difficult to understand mathematics becomes. They are invisible and intangible. To understand a formula, sometimes you can draw and calculate, but more often, you can only imagine in your mind because you can’t draw it.
For example, for the Pythagorean theorem, if you only memorize the formula without understanding its proof process, you will miss a deeper method of calculating the area. It is recommended that if you don’t understand a math concept, start from that point of confusion and spend a lot of time proving it. Understanding one often leads to understanding a whole range of related knowledge.
