Jia Xian pays attention to the systematization of mathematics education in the process of popularizing mathematics knowledge and education? Program? Abstraction and diversification of thinking. From here, we can easily find the flash of his mathematics education thought.
Jia Xian, now aware of his achievements, was the first famous mathematician in Song and Yuan Dynasties. According to the History of Song Dynasty, Jia Shi studied astronomy from Chu Yan, a famous astronomer and mathematician in the early years of Northern Song Dynasty. Calendar calculation of Nine Chapters Arithmetic? Compose? The classic study of island computing is particularly wonderful.
According to records, Jia Xian wrote nine chapters of the Yellow Emperor? Unfortunately, the algorithm to collect and unlock the second volume has been lost. Yang Hui, a famous mathematician in the Southern Song Dynasty, once quoted Jia Xian's Rooting Method and the Method of Increasing, Multiplying and Opening in his book Nine Chapters of Algorithms.
In addition, Jia Xian's "release and unlock method", the perfect "Pythagorean thirteen diagrams of birth and change" and "the method of increasing power and seeking cheap" all show that he abstracted the algorithm? Programming? Mechanization has made important contributions.
Although the information about Jia Xian has not been completely preserved, we can still find some unique mathematical ideas and methods from Yang Hui's Nine Chapters of the Yellow Emperor, including abstract analysis and procedural methods.
In the process of studying Nine Chapters Arithmetic, Jia Xian used abstract analysis, especially in solving the problem of Pythagorean. He first put forward the Pythagorean Thirteen Figures, which has all the relations between Pythagorean chords and their sum and difference, and made an abstract analysis of Pythagorean problems.
It is precisely because Jia Xian mastered this method that he was able to rewrite Nine Chapters of Arithmetic by pure mathematics, leaving a formulaic example for future generations. He also widely used this method in Fine Grass in other chapters such as Equation Skills.
Procedural method mainly refers to the thinking procedure of exploring problems? Process and steps. It is suitable for solving similar problems under the same theoretical system. Jia Xian's "Multiplication and Open Method" and "Multiplication and Cheap Method" particularly embody this method.
Jia Xian formed a fixed procedure in the process of starting the Party. His job is to systematize the prescription procedure? Normalization. Jia Xian's mathematical methodology had a far-reaching influence on mathematicians in Song and Yuan Dynasties. Looking at the "Four Mathematicians of Song and Yuan Dynasties" who made the main achievements in mathematics, we must learn the essence from them.
Jia Xian's "method of increasing, multiplying and opening" initiated the research topic of higher power root, and was improved by Qin's "positive and negative leveling method", which solved the problem of solving the positive root of higher power equation.
In addition, from Ye Li's Tian Shu to Zhu Shijie's Si Yuan Shu, a complete set of equations theory was finally established at the beginning of14th century, which became the most successful subject in the field of mathematics in Song and Yuan Dynasties.
Jiaxian Triangle is called Pascal Triangle in western literature and was rediscovered by French mathematician B Pascal in 1654.
The appearance of Jiaxian Triangle initiated the research direction of higher-order arithmetic progression summation. Zhu Shijie found the "triangle" from every diagonal of the "triangle"? Summation formula of higher-order arithmetic progression such as "scattering star".
In fact, the method of "increasing, multiplying and subtracting" simplifies the calculation program, makes the program more reasonable, and inspires the later calculation and even the improvement of calculation tools.
The mathematical research method initiated by Nine Chapters of Yellow Emperor Jing Cao Shu Jing has been widely used for reference by later mathematicians. In the Qing Dynasty, the academic school "Ganjia School" had annotated or illustrated a number of arithmetic books such as Nine Chapters of the Yellow Emperor when preserving and sorting out its mathematical works.
One of the purposes of ancient scholars writing books is to educate the world. In the process of popularization and education of mathematical knowledge, Jia Xian paid attention to the abstraction of general solution, generalization of knowledge outline, systematization and training of divergent thinking. From here, we can easily find the flash of his mathematics education thought.
Jia Xian attached importance to the abstraction of general solutions. The reason why he did this should be deeply influenced by China's long-standing educational thought that "it is better to teach people to fish than to teach them to fish".
As far as we know, Nine Chapters of the Yellow Emperor was written around 1050. After the book was published, it was widely circulated in the society and gradually replaced Nine Chapters of Arithmetic to some extent. This is also the recognition of his mathematics education thought by the society at that time.
Jia Xian pays attention to the generalization of knowledge outline. After he gave the method of "unlocking the lock immediately", he also put forward the method of "increasing power to get cheaper" and gave the sixth-order Jia Xian triangle to explain the power relationship. When discussing the Pythagorean problem, first discuss the Pythagorean birth and death diagram, and then talk about the solution of the problem, which gives people a clear sense of system.
All his attempts reflect the importance attached to the outline of knowledge. In mathematics education, it is also a good teaching method to pay attention to the generalization of knowledge outline.
According to the available data, Jia Xian has not set foot in the field of Liu Hui's theory of fractions and limits. In addition, the problem of square root discussed in Nine Chapters of Fine Grass in Huangdi Neijing did not involve infinite situations, and he even changed the problem of fractional solution in Nine Chapters of Arithmetic into integer solution. These indications show that his work is based on integer sets.
On this basis, Jia Xian outlined the problems of Pythagorean and square root, and gave the general solutions to many other problems, from which the traces of systematic methods can be faintly seen.
In fact, with Jia Xian's level of mathematical knowledge, it is impossible for him not to be familiar with fractions, and it is impossible for him not to understand Liu Hui's idea of finding the difference, but he has not thoroughly studied the endless problem of prescribing. Therefore, he avoided fractions in his works in order to systematically spread his mathematical knowledge to the world.
This is valuable in the history of ancient mathematics education.
Jia Xian pays attention to the training of divergent thinking. When discussing various problems in Nine Chapters Arithmetic, he did not stick to the ideas and algorithms of his predecessors, but found many new calculation methods. Such as "class division"? "subtraction"? "Art today"? "Comprehensive rate"? Fractionation? Equation technology? "Two shortcomings"? "Pythagoras method" and so on.
It can be seen that Jia Xian not only pays attention to summarizing theoretical research methods, but also devotes himself to the training of divergent thinking, which is of great benefit to the innovation of knowledge.
Nine Chapters of Arithmetic is the most famous mathematical work in China before 1 1 century. During its circulation, many people made notes on it. Three mathematicians have made outstanding contributions to mathematical theory, which is that Liu Hui laid the theoretical foundation? With the improvement of Jia Xian's theory and the basic perfection of Yang Hui's theory, Jia Xian plays a connecting role.
On the other hand, the upsurge of mathematical research in Wei, Jin, Southern and Northern Dynasties was interrupted since the Tang Dynasty. Jia Xian's mathematical methodology stimulated the upsurge of mathematical research in Song and Yuan Dynasties, and he played a role in fueling the situation.
Jia Xian made an abstract analysis of the problems raised in Nine Chapters of Arithmetic to reveal the essence of mathematics. The principle of this method is explained by programming. Outline and sort out the context of knowledge; Pay attention to the systematization of knowledge and avoid paradox. These thinking methods had a far-reaching influence on mathematicians in Song and Yuan Dynasties.
For example, Yang Hui's Detailed Explanation of Nine Chapters Algorithm draws lessons from Jia Xian's abstract and exploration achievements, and reclassifies the problems in the nine chapters; Ye Li's book Measuring the Round Sea Mirror inherits and develops these mathematical methods and establishes a logical deduction system.
Zhu Shijie's "Meeting with Siyuan" also used these thinking methods, making it the pinnacle in the history of ancient mathematics in China. Qin's "Introduction to Mathematical Methods" did not say specific figures, but also imitated Jia Xian, showing the vitality of its methodology.
Of course, these mathematical thinking methods are not original by Jia Xian, but also mathematical writings of past dynasties? Research? The result of accumulation, and Jia Xian refined and passed down.
In a word, the discovery of "Jiaxian Triangle" and the establishment of "multiplication and division" closely related to it played an important role in promoting the peak of classical mathematics in China in the Song and Yuan Dynasties.
Jia Xian