Why does France always produce top mathematicians? Never by talent.

If "like mathematics" solves the entrance problem of top-notch talent training, then the next key is how to identify students with real potential from a wide range of interest groups and transform their potential into stable mathematical ability. France is not through a test to complete the "genius identification",Embedding screening into continuous training, so that students can gradually show their potential in the increasingly difficult mathematical challenges.. Mathematics competition and its follow-up training is an important part of this screening mechanism. The French Mathematical Olympiad is not limited to the examination of knowledge mastery and problem-solving speed, but to observe students' mathematical sensitivity, logical reasoning ability and continuous exploration ability through open and difficult questions. Students with outstanding performance can enter the French Mathematical Olympiad Preparation System (POFM), continue to be challenged through thematic courses, math camps, competitions and intensive training.

Here, there are fewer and fewer fixed templates for the problems faced by students. Proof, discussion and long-term thinking have become important contents of training, and mathematical intuition, abstract thinking and problem-solving skills have been continuously strengthened. It is worth noting that this process undertakes the functions of "training" and "screening" at the same time: students get training in challenges, and then accept a new round of observation and selection in training, thus forming a cycle of "participating in activities-accepting challenges-getting feedback-entering higher-level training. Therefore,"Top-notch" is not a fixed identity formed by a selection, but a state that gradually emerges in the learning process of increasing difficulty.。

After graduating from Senior high school, the university preparatory class (CPGE) become an important interface for the transition of mathematical potential to high-level mathematical ability. CPGE is not an institution that specializes in training mathematicians, but is known for its strict selection requirements and high-intensity courses. For top-notch students in mathematics, its significance lies not only in learning more in-depth and systematic mathematical knowledge, but also in entering a highly concentrated learning environment of excellent students, and forming the ability of rigorous proof, abstract reasoning and dealing with complex problems in intensive training. At the same time, the competition and communication between peers can constantly raise the ability reference, prompting students to continue to break through the original boundaries. From this, the middle school stage mainly answers"Who might have mathematical potential?", CPGE further examined"This potential can withstand high-intensity training”。

The characteristic of France is that it does not separate the two, but through the gradual connection of competitions, professional mathematics community and elite education, so that talent identification and ability training are nested in each other. Students do not have only one fixed channel, but can enter a higher-level training system from different entrances such as school curriculum, math clubs, competitions, etc., and then gradually converge in continuous challenges.

After entering higher education, the focus of training top-notch mathematics talents in France began to shift from strengthening learning ability to strengthening research ability. In order to higher normal college (US) as the representative of the elite institutions, it is an important platform to complete this transformation.

Image source:Paris Normal University Library

This kind of "research" is first reflected in the change of training mode. Taking the Department of Mathematics and Application (DMA) of Paris Normal University as an example, students do not study according to the unified curriculum template,Under the guidance of the instructor, choose advanced courses according to personal interests and development direction, and can extend the study to computer science, physics and other related fields. With the deepening of learning, thematic discussions, working groups, research internships, etc. gradually increase, students need to read the literature, report research progress and participate in peer discussions. The focus of school training has gradually shifted to "whether students can learn and explore by themselves". Students can contact real mathematical problems through research papers, special projects and scientific research practice at the undergraduate and master levels, and learn to consult literature, understand existing results, form arguments and explore their own ideas under the guidance of their tutors. The evaluation criteria change accordingly: students are no longer faced with exercises that have preset answers, but whether they can continue to advance a question in uncertainty.

An important reason why this change can happen is that there is no obvious institutional barrier between the French elite mathematics education and scientific research system. The Department of Mathematics of ENS and the French National Center for Scientific Research (CNRS) have achieved deep integration through joint research laboratories. University teachers, CNRS researchers and postdocs jointly create an academic environment around students. As a result, students can enter the real mathematical research community early, observing how mathematicians ask questions, evaluate problems and carry out research through lectures, discussion classes, research projects and research internships.

For students who fail to enter elite universities such as higher normal colleges, universities are also an important position for the training of top-notch mathematics talents.. After the students enter the master and doctoral stage, the research-based training model is further deepened. Taking the University of Paris Thackeray as an example, its Jacques Hadamard mathematics master's degree and other programs further lead students with outstanding mathematical foundation to doctoral research through high-level courses, research practice and inter-institutional research network. At the doctoral stage, the transformation from "receiving research training" to "independently producing knowledge" is finally completed, and young researchers begin to form their own research results around original problems.

Therefore, from CPGE to ENS and the postgraduate stage of universities, what happens is not a simple increase in the amount of mathematical knowledge, but a change in mathematical cognitive style and academic identity: CPGE trains students more about how to solve difficult problems, while research training further trains them how to find problems worth solving. An excellent mathematics learner thus gradually enters the mathematical community and becomes a young researcher involved in knowledge production.

After graduation, the key to talent growth gradually shifted from "how to cultivate" to "how to make creative research continue to happen". An important feature of the French mathematics system is the construction of a scientific research ecology with CNRS as the core and the participation of universities, elite schools and research institutions, which provides young researchers with long-term growth time and broad growth space.

The first is a stable research time and career space.. CNRS recruits and absorbs doctoral graduates into the vocational research system through national competition, and relies on joint research laboratories built with universities and other institutions to provide researchers with an environment for continuous basic research. In recent years, the implementation of the "young professor seat" and other systems has further provided researchers in the early career with several years of scientific research support and relatively stable development expectations. For mathematical research that is highly dependent on long-term thinking, the significance of this system is not only to provide positions and funding, but also to give young researchers enough time to form their own research directions without being prematurely driven by short-term results.

followed by an active research community. French mathematical research does not take isolated individuals or single institutions as the basic unit, but relies on CNRS-University Joint Laboratory, thematic research network and so on to form an inter-institutional academic community. Young researchers continue to exchange ideas with their peers through discussion classes, academic reports, thematic conferences and collaborative research, and gradually establish their academic direction and reputation in the process of presenting results, responding to questions and solving problems together. Talent growth is thus no longer just a one-way training of students by tutors, but a process in which the entire academic community continues to participate.

At the same time, international flows further expand the boundaries of this community.. The National Institute of Mathematics and its Interaction (INSMI) under CNRS promotes transnational cooperation through international research laboratories, international research networks and other mechanisms, while platforms such as the Henry Poincare Institute (IHP) in Paris and the International Mathematical Conference Center (CIRM) in Marseille bring together mathematicians from different countries to carry out intensive exchanges on specific issues. As a result, young researchers can not only understand international cutting-edge theories, but also directly participate in the problem discussion and knowledge production of the global mathematical community, forming a more original research direction in the continuous collision.

It can be seen that the training of French mathematics talents does not stop at "cultivating a person who can do research", but further creates conditions so that young researchers have time to explore a problem in depth and have the opportunity to discuss with their peers. Opportunity to enter the international frontier.As the research direction matures, they become mentors, team leaders and academic community builders, and begin to train the next generation of mathematical talents.。

From the experience of France, top-notch mathematics talents are not "made" by a certain selection or an elite school,In the long-term, continuous education and research ecology gradually grow up. Its outstanding feature is that it does not simply equate talent training with "selecting the smartest students as soon as possible", but connects interest discovery, dynamic screening, high-intensity training, research training and scientific research career development: expand students' opportunities to contact and explore mathematics as much as possible in the early stage, and gradually show students' mathematical potential through continuous challenges in the medium term. After entering higher education, students will be promoted to change from learning knowledge to creating knowledge, after becoming a researcher, it is given enough time, community support and international exchange space to enable creativity to continue to grow.

This experience tells us,The cultivation of top-notch talents requires not only "selection", but also "support"We should not only pay attention to a small number of top students who have emerged, but also reserve the entrance for potential talents who have not yet been discovered. We should not only pay attention to the curriculum and training in the training stage, but also pay attention to the institutional space for long-term development after talents stand out. In this sense, the most worthy of reference for the training of French mathematics talents may not be a specific system, but the "long-termism" that runs through the whole process of talent growth ".