The importance of mathematics is widely recognized. However, because mathematics is a science that studies only the quantitative relationships and spatial forms of its objects, disregarding all other material properties, it often appears in a highly abstract form. This high degree of abstraction does not mean that mathematics is isolated from the real world; on the contrary, it provides broader possibilities for the practical application of mathematics. However, for mathematics to be applied and demonstrate its powerful vitality, it must find a way to build a bridge between the real-world problems it faces and mathematics itself. This means transforming real-world problems into corresponding mathematical problems, then analyzing and calculating these mathematical problems, and finally applying the solutions back to reality to see if they can effectively answer the original real-world problem. This entire process, especially the first step, is called mathematical modeling—establishing a mathematical model for the real-world problem under investigation. Undoubtedly, mathematical modeling is a crucial bridge connecting mathematics and its applications, and an essential path for mathematics to achieve practical application.
Furthermore, mathematical modeling plays a crucial role in related disciplines and applications. In the 3rd century BC, the ancient Greek mathematician Euclid, building upon the achievements of his predecessors, established Euclidean geometry, a mathematical model of the spatial forms of the real world. This model is highly effective and, despite various important developments, continues to play a vital role today. The three laws of planetary motion, summarized by the German astronomer Kepler based on Tycho Brahe's extensive astronomical observations, were later rigorously proven by Newton using the law of universal gravitation (inversely proportional to the square of the distance), based on the principles of Newtonian mechanics. This is another brilliant example of the success of mathematical modeling. The fundamental differential equations in some important fields of mechanics and physics (such as particle mechanics, electrodynamics, fluid mechanics, and quantum mechanics) are also mathematical models that capture the essence of the discipline, forming the core content and basic theoretical framework of the subject, containing all its important results and possible applications.
In the 21st century, the application scope of mathematics has expanded unprecedentedly, extending from traditional fields such as mechanics and physics to various disciplines including chemistry, biology, economics, finance, information technology, materials science, environment, and energy, as well as various high-tech and even social fields. Because the laws governing many new fields are still being explored, mathematical modeling remains a formidable challenge. This not only further highlights the importance of mathematical modeling but also makes it a crucial breakthrough and core component in the development of modern applied mathematics.
Mathematical modeling is not only an essential path for mathematics to be applied, but also a sure way to enlighten the mathematical mind. Mathematics education is essentially a form of holistic education; it should enable students to grasp the spirit and methodology of mathematics, master the essence of the discipline, consciously embrace the influence of mathematical culture, and make mathematics a readily available and invaluable tool throughout their lives. Since the introduction of the "Mathematical Modeling" course in China in 1982 and the annual National Undergraduate Mathematical Modeling Contest in 1992, over the past 30 years, education and competitions related to mathematical modeling have broken down the self-contained and closed nature of traditional mathematics courses, opening a channel and providing an effective way to connect mathematics with the outside world in the teaching process. Through participation in mathematical modeling learning and practice, students personally engage in attempts to apply mathematics to real-world situations, and personally participate in the process of discovery and creation, gaining valuable experiences and firsthand insights that were previously unattainable in the classroom and from books. These activities have inspired their mathematical minds, encouraging them to better apply, appreciate, understand, and love mathematics. They integrate the cultivation and assessment of knowledge, abilities, and qualities, and are open to university students in almost all majors, effectively promoting the cultivation of innovative and outstanding talents. This is the longest-running, largest-scale, and most successful mathematics teaching reform practice in China in recent years, receiving widespread recognition, enthusiastic welcome, and strong support from all sectors of society and teachers and students. The continued flourishing of mathematical modeling education and competitions also fully reflects the importance of mathematical modeling.