(1) Ancient People's Understanding of Mathematics

Aristotle in ancient Greece believed that mathematics was a science that studied quantity. And said that "number is discrete" and "line is continuous". The study of numbers and their properties (such as parity, symmetry and proportional relationship) is called arithmetic, and the study of quantities and their properties (such as symmetry, intersection and parallelism) is called geometry.

(2)1People's understanding of mathematics before the 9th century.

The history of mathematics shows that before19th century, the main achievements of classical mathematics were arithmetic, geometry, algebra and calculus. These mathematical disciplines study the spatial form and quantitative relationship of objective things. In this regard, Engels once summarized it as: "The research object of pure mathematics is the spatial form and quantitative relationship of the real world."

(3) Modern people's understanding of mathematics.

The Bourbaki school believes that "mathematics is the science of studying abstract structures." They look at mathematics from the viewpoint of structure, and think that the most common and basic mathematical structures are algebraic structure, sequential structure and topological structure, which are three parent structures, and there are many various substructures, which together with some substructures constitute a branch structure of mathematics;

Alexander Love, a famous mathematician in the Soviet Union, pointed out in his book Mathematics-Its Content, Method and Significance: "Mathematics takes pure formal relations and forms as its object." ;

Chinese mathematician Ding believes that "the research object of mathematics is the objective and logically possible mathematical relationship and structural relationship." ;

There are still many mathematicians who believe that Engels' view of mathematical objects is still applicable to modern mathematics as long as the understanding of quantitative relations and spatial forms is expanded.

At present, the Mathematics Teaching Syllabus for Full-time Senior High Schools (Experimental Revised Edition) ignores the "real world" defined by Engels when it comes to the object of mathematics, that is, "mathematics is a science that studies the relationship between spatial form and quantity".

These viewpoints summarize the objects of mathematics from different aspects and are not contradictory in essence. Because people have not found a more accurate and acceptable statement about mathematics,

Here we temporarily use the current "Full-time Senior High School Mathematics Teaching Syllabus" (experimental revised edition): "Mathematics is a subject that studies the relationship between spatial form and quantity.

Talking about the novelty of mathematics?

[Author: Li Lingzhi of Lize Middle School reposted from: original hits on this site: 443 Update time: 2006- 10-8 Article entrance: Lize Middle School]

On the understanding of the history of mathematics

Lize Middle School affiliated to Capital Normal University

Li Lingzhi

A teacher said with emotion: although we have taught mathematics for so many years, we really don't know much about the history of mathematics. In the future, we should learn more about the history of mathematics through various channels, enrich our "mathematical knowledge base" and let students feel more about the inner charm of mathematics in mathematics classes.

First, the significance of learning the history of mathematics

Studying the history of mathematics is of great significance to every mathematician, especially to us, the disseminators of mathematical knowledge. I think the significance of studying the history of mathematics mainly includes the following three points: 1, the scientific significance of the history of mathematics.

Every science has its history of development. As a historical science, it is both historic and realistic. Its reality is first manifested in the continuity of scientific concepts and methods. Today's scientific research is to some extent the deepening and development of scientific tradition in history, or the solution of scientific problems in history, so we can't separate the relationship between scientific reality and scientific history. Mathematical science has a long history. Compared with natural science, mathematics is an accumulative science, and its concepts and methods are more continuous. For example, the decimal notation and the four arithmetic rules formed in ancient civilization have been used to this day. Historical issues such as Fermat's conjecture and Goldbach's conjecture have long been hot topics in the field of modern number theory, and materials of mathematical tradition and history can be developed in practical mathematical research. Many famous mathematicians at home and abroad have profound cultivation or research on the history of mathematics, and are good at drawing nutrients from historical materials, making the past serve the present and bringing forth the new. Wu Wenjun, a famous mathematician in China, made outstanding achievements in the field of topology research in his early years. In the 1970s, he began to study the history of Chinese mathematics, which opened up a new situation in the research theory and method of the history of Chinese mathematics. Especially inspired by China's traditional thoughts of mathematical mechanization, he established a mathematical mechanization method for mechanical proof of geometric theorems, which was called "Wu Fa" in history. His works are worthy of being a model of making the past serve the present and revitalizing national culture. The reality of the history of science also lies in providing experience and lessons for our scientific research today, making us clear the direction of scientific research, avoiding detours or mistakes, providing a basis for today's scientific and technological development decisions, and also providing a basis for us to foresee the future of science. If we know more about the history of mathematics, we won't have such absurd things as drawing the third part of the solution angle and proving the four-color theorem, and we will also avoid wasting time and energy on Fermat's last theorem and other issues. At the same time, summing up the experience and lessons in the history of mathematics development in China is beneficial to the development of mathematics in China today. 2. Cultural significance of the history of mathematics.

An American historian of mathematics once said: "The general characteristics of an era are closely related to the mathematical activities of this era to a great extent. This relationship is particularly evident in our time. " Mathematics is not only a method, an art or a language, but also a rich knowledge system, which is very useful to natural scientists, social scientists, philosophers, logicians and artists and influences the theories of politicians and theologians. Mathematics has widely influenced human life and thought, and is the main force to form modern culture. Therefore, the history of mathematics reflects the history of human culture from one side and is the most important part of the history of human civilization. Many historians understand the characteristics and value orientation of other major ancient cultures through the mirror of mathematics. Mathematicians in ancient Greece (600 BC-300 BC) emphasized strict reasoning and the conclusions drawn from it, so they did not care about the practicality of these achievements, but educated people to make abstract reasoning and inspired people to pursue ideals and beauty. Through the investigation of the history of mathematics in Greece, it is very easy to understand why ancient Greece had beautiful literature, extremely rational philosophy and idealized architecture and sculpture that could not be surpassed by later generations. The history of Roman mathematics tells us that Roman culture is foreign, and the Romans lack originality and pay attention to practicality. 3. Educational significance of the history of mathematics.

When we have studied the history of mathematics, we will naturally feel that the development of mathematics is illogical, or that the actual situation of mathematics development is very inconsistent with the mathematics textbooks we have learned today. The mathematics content we learn in middle schools today basically belongs to the elementary mathematics knowledge before calculus in17th century, while most of the contents of the mathematics department in universities are advanced mathematics in17th and18th century. These mathematics textbooks have been repeatedly tested and compiled under the guidance of the principle of combining science with educational requirements. They are knowledge systems that compile historical mathematical data according to certain logical structure and learning requirements, and inevitably abandon the actual background, knowledge background, evolution process and various factors that lead to the evolution of many mathematical concepts and methods. Therefore, it is difficult to obtain the original appearance and panorama of mathematics only by studying mathematics textbooks. At the same time, it ignores those mathematical materials and methods that have been eliminated by history but may be useful to real science, and the best way to make up for this deficiency is through the study of mathematical history. In the eyes of ordinary students, mathematics is a boring subject, so many of them regard it as a road to fear. To some extent, this is because our math textbooks often teach some rigid and unchangeable math content. If the history of mathematics is infiltrated into mathematics teaching to make mathematics alive, it will stimulate students' interest in learning and help deepen their understanding and understanding of mathematical concepts, methods and principles. The history of science is an interdisciplinary subject of arts and sciences. Judging from today's education situation, the gap between arts and sciences has made the talents trained by our education increasingly unable to adapt to today's modern society with high penetration of natural science and social science. It is precisely because of the interdisciplinary nature of the history of science that it can show the role of communicating arts and sciences. Through the study of the history of mathematics, students in the department of mathematics can receive the training of mathematics major and get the cultivation of humanistic quality, while students in liberal arts or other majors can learn the general situation of mathematics and get the cultivation of mathematics and physics through the study of the history of mathematics. The achievements and moral character of mathematicians in history will also play a very important role in the personality cultivation of teenagers. Mathematics has a long history in China. /kloc-Before the 4th century, it was the most developed country in the world. Many outstanding mathematicians have appeared and made many brilliant achievements. Its long history, calculation-centered, programmed and mechanized algorithmic mathematical model and the axiomatic mathematical model characterized by geometric theorem deduction and reasoning in ancient Greece reflect each other and alternately influence the development of world mathematics. Due to various complicated reasons, China became a mathematical superpower after16th century. After a long and difficult development process, it gradually merged into the trend of modern mathematics. Due to educational mistakes, under the influence of modern mathematical civilization, we often forget our ancestors and know nothing about the traditional science of our motherland. The history of mathematics can help students understand the brilliant achievements of ancient mathematics in China, the reasons for the backwardness of modern mathematics in China, the present situation of modern mathematics research in China and the gap with developed countries, thus stimulating students' patriotic enthusiasm and revitalizing national science.

Second, the role of learning the history of mathematics in mathematics teaching 1. Investigate history and carry out patriotic education.

China has a glorious history of mathematics. Many outstanding mathematical achievements in ancient times had an important impact on ancient human civilization. There are many vivid materials in middle school mathematics textbooks. Digging deep into the patriotic education factors in textbooks, introducing the history of China's mathematics development, introducing the outstanding achievements of ancient China scientists and introducing the great contribution of modern China people to the development of mathematics can stimulate students' strong national pride, self-confidence, pride and patriotic enthusiasm. For example, when teaching the formula for calculating the area of simple geometric figures, a series of masterpieces handed down from ancient times in the history of mathematics in China, represented by "Nine Chapters of Arithmetic", are appropriately introduced to students; When teaching negative numbers, introduce the appearance of negative numbers and the origin of their use in mathematics. Zu Chongzhi, a famous mathematician in the Northern and Southern Dynasties, calculated 3. 14 15926 when he was teaching Yuan.