Summary of Key Points
Recently, there have been two contrasting events in the mathematical community: First, Wang Hong and Deng Yu, alumni of the class of 2007 from Peking University, are both considered to have a chance of winning the Fields Medal (the Nobel Prize in mathematics), which would create history for Chinese mathematicians. Second, there are rumors that Tsinghua University has expelled a group of "mathematical geniuses" who entered its preparatory program through special selection processes. These students failed exams (with an average score of only 110 on the Beijing college entrance examination, lower than that of ordinary high school students), leading to speculation about the role of rote practice in exam preparation versus a passion for genuine exploration in mathematics education, as well as the kind of environment necessary to nurture top-tier talent.
1. How prestigious is the Fields Medal? Why could two Peking University alumni make history?
The Fields Medal is considered the Nobel Prize in mathematics, awarded every four years to mathematicians under the age of 40 and carries great prestige. The simultaneous nomination of Wang Hong and Deng Yu marks the first time Chinese mathematicians have achieved such a double honor.
- Wang Hong's story: She entered Peking University at the age of 16 to study earth sciences but was fascinated by mathematical formulas during her freshman year, prompting her to switch to the mathematics department. Initially, her grades were poor, but she avoided rote practice and instead spent months pondering a single problem, trying out all possible approaches. She eventually solved the "Hanging Valley Conjecture," which had stumped mathematicians for 107 years. Her approach involved exploring various possibilities to understand how a needle can cover the maximum area in three-dimensional space.
- Deng Yu's background: After two years at Peking University, he transferred to MIT and earned his Ph.D. from Princeton, later becoming a professor at the University of Chicago.
Both of them did not have an innate advantage; instead, they used their passion and perseverance to pursue their goals, delaying immediate gratification (Wang Hong said, "I'm not a genius; I just persist."
2. The controversy surrounding Tsinghua's preparatory program: Why can't these geniuses outperform ordinary high school students?
The preparatory program at Tsinghua allows students to bypass the regular college entrance examination with advanced tests in calculus and linear algebra (with only about a hundred candidates selected nationwide). After a four-month preparatory period, those who pass are admitted. However, many of these highly talented students failed their final exams, scoring below average on the Beijing college entrance examination.
- The issue: These students may have been overtrained with too many targeted practice problems, leading to a lack of flexibility when faced with unfamiliar questions. For example, parents of the expelled students claimed that their children could only solve the types of problems they had practiced and struggled with new formats.
3. What is "overfitting"? How does rote practice turn geniuses into machines?
The article uses the concept of "overfitting" in artificial intelligence to explain this phenomenon:
- Imagine training an AI to recognize cats by showing it 10,000 photos of cats on a red carpet. The AI learns that "red carpet + animal = cat," but fails to recognize cats on a sofa or even mistakes a ball for a cat—this is overfitting; it has memorized specific answers without understanding the underlying principles.
The students in the preparatory program, having practiced only the types of problems used in the selection process, lack the ability to apply their knowledge flexibly.
4. Two approaches to learning mathematics: "Walking on a tightrope" vs. "Weaving a net"
A teacher from Tsinghua's preparatory program compared these two methods:
- Walking on a tightrope: Focusing on exam preparation, which offers a quick shortcut but has a high risk of failure (like the students who struggled with the more advanced problems in the preparatory program).
- Weaving a net: A learning approach driven by passion, where knowledge is interconnected, allowing learners to derive new insights even if they forget certain formulas (as Wang Hong did, eventually finding her own methods through exploration).
5. What kind of environment fosters top-tier mathematicians?
Wang Hong's success highlights the importance of an environment that encourages experimentation and delayed gratification:
- Her path: After graduation, she went to a prestigious French research institute, where many Fields Medalists have studied, and plans to establish a mathematics lab in her hometown to inspire young students.
- Reflection: While our basic education system can produce promising candidates, it often lacks the space for independent exploration. The rushed nature of the preparatory program at Tsinghua, with its frequent exams, hinders the development of true mathematical talent.
In summary, these two events highlight that mathematics is not about rote practice; it requires a combination of passion, exploration, and perseverance. To cultivate top-tier mathematicians, we need to provide an environment that supports experimentation and allows for a gradual mastery of knowledge, rather than forcing them to follow a strict, error-prone path.
(Note: The Tsinghua expulsion has not been officially confirmed; the information in this article is based on rumors and should be verified by official sources.)