I am not good at math, is there any hope for my child’s math?

You may have heard many jokes that make fun of mathematics, such as "Life may deceive you, but mathematics will not, because if you don't know mathematics, you don't know it", and "The Chinese meaning of the English word aftermath is trauma", etc. Many people even become afraid of the word "math".

People often use the term "math geniuses" to refer to mathematicians. So what is the relationship between math learning and talent?

01

From the Seven Bridges Problem to the One-Stroke Problem

See how math geniuses solve math problems?
There was once a very famous problem in history called the " Seven Bridges Problem ".

In the early 18th century, in Königsberg, Prussia, there was a river running through it, with two small islands on the river and seven bridges connecting the two islands to the riverbank. Someone asked the following question: How can a walker walk across all seven bridges at once without repeating or missing any bridges and finally return to the starting point?

Image credit: Alamy Stock Photo

It is not difficult to find that this problem is actually a one-stroke drawing problem. After the Seven Bridges Problem was proposed, many people were very interested in it and tried to draw pictures to exhaust all possibilities, but it has not been solved for a long time.

In 1736, 29-year-old Euler submitted a paper entitled "The Seven Bridges of Königsberg" to the St. Petersburg Academy of Sciences, in which he directly gave the definition of odd and even points, and proposed the inference that a curve can only draw two odd points, thus simplifying this type of drawing problem to counting the number of odd points .

Euler transformed the two banks and the island into points, and the bridges into lines, and by counting the number of singular points, he concluded that it was impossible to walk across all seven bridges at once.

Image source: Emmett Peng

While solving the problem, Euler created a new branch of mathematics - graph theory and geometric topology , which also started a new journey in the history of mathematics. Compared with the exhaustive method, Euler's way of thinking about problems is quite like a dimensionality reduction attack.

As a mathematician, Euler's achievements go far beyond this. He was one of the most outstanding figures in mathematics in the 18th century and made outstanding contributions in many fields. It is worth mentioning that Euler himself had a superb memory and mental arithmetic skills . This enabled him to calculate most results through mental arithmetic even after he became blind.

Swiss mathematician Leonhard Euler

Image source: Science Popularization China

In the long history of the development of mathematics, there are many mathematical geniuses like Euler, including but not limited to Gauss, Galois, etc. I think everyone can also name the deeds of several well-known mathematicians.

02From the International Mathematical Olympiad to the Fields Medal , what is the relationship between mathematical achievement and mathematical talent? Having mathematical talent is naturally a positive bonus for achieving success in mathematics.

I believe everyone is familiar with the term "Mathematical Olympiad". The International Mathematical Olympiad is the highest level of mathematics competition, and its contestants are the best among middle school students. Participants who can win awards in the International Mathematical Olympiad must have high mathematical talent. In the field of mathematics, the Fields Medal is one of the highest international awards. A major feature of the Fields Medal is that the winner must be under 40 years old before New Year's Day of that year, so they are well-deserved to be called mathematical geniuses.

The Fields Medal, the "Nobel Prize in Mathematics", has a relief portrait of Archimedes on the front of the medal

Image source: ScienceNet

The author was once curious about the relationship between "International Mathematical Olympiad winners" and "Fields Medal winners", so he looked up relevant information and found that among the 26 winners of the seven Fields Medals from 1998 to 2022, 12 were winners of the International Mathematical Olympiad, accounting for nearly half .

Among them, Chinese-American mathematician Terence Tao , winner of the 2006 Fields Medal, won a bronze medal, a silver medal, and a gold medal in the International Mathematical Olympiad in the three years from 1986 to 1988; Peter Scholze , winner of the 2018 Fields Medal, won three gold medals and one silver medal in the International Mathematical Olympiad between 2004 and 2007.

Chinese-American mathematician and Fields Medal winner Terence Tao (left); German mathematician and Fields Medal winner Peter Scholze (right)

Image source: Chinese Mathematical Society

It can be seen that the mathematical talent revealed in middle school has a close connection with becoming a top mathematician.
It is true that most of us are far from being top mathematicians, but does this mean that we are not good at mathematics? Not necessarily.

03

From Fermat to Garfield

What are the necessary conditions for learning mathematics well?

It takes a high level of talent to study cutting-edge mathematical problems, and when it comes to learning mathematics itself, talent is certainly better, but there are many other influencing factors.

Under normal circumstances, learning mathematics well also requires diligence. Mathematical geniuses are often considered to have innate talent, but talent is only part of success. Many mathematical geniuses have also invested a lot of time and effort in exploring mathematics. The author believes that talent determines the upper limit, while diligence determines the lower limit .

In addition, learning mathematics is inseparable from interest and love for mathematics. Mathematicians often have a deep love and interest in mathematics, which drives them to constantly explore the mysteries of the mathematical world. For us, passion for mathematics should also be the source of motivation to learn mathematics well.

Throughout history, there have been many "amateur" mathematicians, and the famous Fermat is known as the "king of amateur mathematicians." Fermat has made considerable achievements in the fields of number theory and analytic geometry, but in fact he had a full-time job as a lawyer.

Fermat, the most "slacking off" civil servant on earth

Image source: Tongji Mathematics Youth

Another interesting story is that when the 20th President of the United States, Garfield , was a member of Congress in 1876, he proposed a unique proof of the Pythagorean theorem, which has also been passed down as a good story in the history of mathematics. There are many similar stories, and behind these achievements is a passion for mathematics.

Having a high talent for mathematics may be a sufficient condition for learning mathematics well, but it is not a necessary condition. In the process of learning mathematics, if we have less utilitarianism and more interest, I believe everyone can enjoy the fun brought by mathematics.

Image source: Chinese Mathematical Society

Author: Wang Yuchao, PhD in Basic Mathematics, teacher at the Department of Mathematics, School of Science, Shanghai University

Editor: Little Dandelion