2. Connotation of the history of mathematics.
To fully understand a thing, we must understand its ins and outs, and to learn mathematics, we must ask about the development of mathematics. "Studying the history and present situation of this subject is an appropriate way for students to predict the future of mathematics." To quote Henri Poincare, a famous French mathematician, that is to say, if we only emphasize the mastery of knowledge and don't know the development history of this knowledge, then for these students, what they have learned is only a fragment of mathematics, and they can't really understand the subject of mathematics, but the history of mathematics can show us the overall face of knowledge and let us better understand the past, present and future of mathematics.
As a science to study the emergence, development and laws of this subject, the history of mathematics is not only as simple as historical knowledge, but also can be traced back to the connotation of mathematics and the evolution and development of logical thinking mode. In addition, the influence of mathematical development on human civilization over the past 5,000 years and its pivotal position in human history are also studied. Some people simply think that the study of the history of mathematics is only to find out what knowledge was put forward by which mathematician in which year, and what knowledge humans know and don't know so far. Undoubtedly, this is one of the tasks to be studied in the history of mathematics, and it is also the most basic work. But the more important purpose of studying the history of mathematics is to let teachers and students stand on the achievements of modern mathematics, clarify the development direction and law of the subject from the source, and recognize its logical thinking mode, so as to better understand mathematics and learn mathematics in essence.
3. The role of mathematics history in middle school mathematics teaching.
Under the tide of reform under the new curriculum standard, mathematics textbooks in middle schools have correspondingly increased a lot of knowledge about the history of mathematics. So, what role does the history of mathematics play in middle school mathematics teaching? As a prospective teacher who is about to leave school and engage in mathematics teaching, I think it has the following functions:
3. 1 The history of mathematics can stimulate students' interest in learning mathematics.
The new curriculum standard emphasizes that teachers should not only pay attention to the process and methods in the teaching process, but also pay attention to students' emotions and attitudes. Only in this way can students have a strong interest in learning. In the eyes of many students, mathematics is a boring subject, which is neither as beautiful as Chinese nor as practical as English, making many people uninterested in learning. However, mathematics is indispensable in human civilization. This is a very logical and abstract theme. If we simply talk about mathematics knowledge and don't pay attention to cultivating students' interest in mathematics, then students will only learn passively and their initiative in learning will be restrained. The history of mathematics is of great help to stimulate students' interest in learning mathematics. Infiltrating the history of mathematics in mathematics classroom teaching can make mathematics teaching active, which is not only conducive to deepening the learning effect, but also can stimulate and improve students' interest in learning mathematics.
At the beginning of the class, telling mathematicians' stories according to the teaching content can arouse students' strong interest and shift their attention from extracurricular activities to mathematics teaching. This is a good way to create the best classroom situation and pave the way for classroom teaching. Moreover, teachers can broaden students' horizons by telling mathematical allusions, let them know that there is such a story behind these seemingly boring knowledge, and thus become interested in what they have learned. For example, when talking about the sum of the first n items in the series, tell students the story that Gauss Primary School was fined at the beginning of class because of the sum of the first 100 positive integers, so that students' thinking will soon be attracted to the classroom. In addition, the introduction of famous historical topics by teachers in the classroom has also played a role in stimulating students' interest. Many famous historical topics are related to mathematicians. When students think about problems, they will inadvertently think of them. Many great mathematicians have thought about it and they will feel a challenge. Many great mathematicians have also thought about the topic they are thinking about now. I don't know if they are confused like me. Even if they can't think of it, they will have a strong interest in the topic.
3.2 The history of mathematics can deepen students' understanding of mathematical knowledge.
Middle school students' mathematics textbooks are limited by some restrictive factors. Although the knowledge taught is systematic, students still can't have a clear and detailed understanding of the ins and outs of knowledge. We can make use of the process law of human cognition in the history of mathematics to sort out the backbone of knowledge vertically, so that the context of knowledge in students' minds is clearer, which is conducive to students' profound understanding and memory of knowledge. The history of mathematics can make students accept new knowledge more easily. When they first come into contact with algebra and replace specific numbers and numbers with letters for the first time, they are often confused. I don't know why. At this time, if teachers want to change this situation, they can tell students relevant mathematical historical materials in class to help students sort out and understand the mathematical knowledge they have learned. Mathematics has a long history of development, but what students learn today is all obtained through indirect learning. The difficulties experienced by mathematicians in the past are exactly the obstacles that students are experiencing now. Because the process of knowledge generation is very similar to that of students' indirect learning, the teaching of mathematical history can help students better understand mathematical knowledge. Generally speaking, mathematical knowledge is closely related. Through the history of mathematics, the knowledge learned in the mind is sorted out, so that students can better establish the relationship between knowledge points, disciplines, study and life in their minds, and pave the way for a deeper understanding of mathematics.
In the history of mathematics, the emergence of irrational numbers triggered the first mathematical crisis. For a long time, people are psychologically unwilling to accept this fact. It is not easy for students to learn this irrational number that once caused a stir. A middle school in Shanxi has done a survey. For the knowledge about irrational numbers, 70% of students only know how to do problems, and they don't have a deep understanding of the concept of irrational numbers, which is bound to have a certain impact on later learning. Looking up the relevant mathematical historical materials, we find that the process of people discovering and understanding irrational numbers in the history of mathematics is long, and many mistakes have been made in this process, so that we can understand it well. It is not surprising that students encounter difficulties in learning this concept, which is only a "reproduction" of history. Therefore, teachers can tell students more about the history of irrational numbers in class, which will help students understand and accept this knowledge.
3.3 The history of mathematics helps students to master mathematical thinking methods.
Mathematics is a special subject, which is unique in that it has an extremely strict logical form of thinking. The reason why we want to learn mathematics is that we hope to exercise our brains in the process of learning mathematics, form an accurate and meticulous logical thinking mode, and exercise to improve our creativity. Facts have proved that the history of mathematics has played an indelible role in the realization of this educational goal. Nowadays, middle school mathematics textbooks present students with more systematic and seamless knowledge, and the language is very concise. Basically, they are arranged in fixed forms such as definition, theorem, proof, reasoning and example exercises. In the process of learning, students simply accept this knowledge and lack the real mathematical thinking process. Due to the limitation of students' cognitive level, it is easy to produce incorrect views and ideas. Although we can accept a large amount of knowledge quickly and conveniently, it is easy for students to think that the process of learning mathematical knowledge is "definition-getting property theorem-doing problems", and the fact is systematic, but we can't let students clearly realize that knowledge is gradually mature through discovering problems, putting forward assumptions, demonstrating assumptions, drawing conclusions and perfecting, which is not conducive to the formation of students' correct mathematical thinking methods. However, the history of mathematics can do this. The history of mathematics presents students with not only clear mathematical knowledge, but also the creative process of imparting corresponding knowledge, which makes students have a clearer understanding of the generation of mathematical knowledge. Through the history of mathematics, we can realize the origin and characteristics of mathematics. From this perspective, under the guidance of the history of mathematics, a two-way classroom atmosphere of exploration and research can be created between teachers and students.
There are many such examples. For example, when we talk about the idea of combining numbers with shapes, we can first tell students that there are many long-standing unsolved problems in geometry, such as three times, bisection of any angle, turning a circle into a square and so on. Until the second half of the17th century, the French mathematician Descartes established the corresponding relationship between points and numbers, curves and equations with coordinates as a bridge, and studied geometric problems with algebraic methods, thus creating explanatory geometry, which has been in use ever since. For another example, Newton and Leibniz founded calculus to solve many scientific problems on the basis of the ideological achievements of ancient mathematicians in studying integral calculus.
3.4 The history of mathematics can cultivate students' spirit of exploration.
Generally speaking, the mathematics textbooks that students study present students with systematic and ready-made knowledge, which fails to reflect the hardships of mathematicians who advance wave after wave and cut corners to acquire mathematical knowledge. The arduous and long road experienced by mathematicians seems to be just a form for students. However, the prosperity of mathematics today depends on generations of mathematicians who have the courage to explore and struggle. By studying the history of mathematics, students can understand this truth, know how these mathematicians struggled hard, how to overcome all difficulties, how to accumulate knowledge bit by bit, and give the latecomers a better knowledge environment. They will find that the difficulties they have experienced in learning mathematics at present are negligible, so they will not be knocked down by setbacks encountered in the learning process. In addition, through the history of mathematics, students will find that many famous mathematicians have made mistakes that seem ridiculous now, and mathematicians will make mistakes like them, so as to correctly treat their mistakes in the process of learning mathematics and establish their self-confidence in learning mathematics.
Taking the calculation of pi ∏ as an example, many people at home and abroad have devoted themselves to the research and calculation of pi. In order to calculate a better approximation of pi, countless mathematicians have devoted their lives and efforts to this mysterious number. Before19th century, the calculation of pi progressed very slowly. After19th century, the world record for calculating ∏ was frequently innovated. Ludolph Van Ceulen of Germany spent almost his whole life calculating the inscribed regular 262-sided polygon of a circle by classical method, and got the 35-bit precision value ∏ in 1609, so that ∏ is called Ludolph number in Germany; It took William Shanks of England 15 years to calculate the 707th decimal of pi 1874, and engraved it on the tombstone as a lifelong honor. Unfortunately, later generations found him wrong from the 528th place. Although there was a computer later, people were still interested in pi, because mathematicians thought that the study of ∏ could show that human knowledge was endless. When teaching pi, it is of positive significance to tell students appropriate historical knowledge, which is helpful to cultivate students' exploration spirit of facing difficulties. Mathematicians of all ages have made great contributions to the Babel of Mathematics by cutting through difficulties. Their lofty ideals, firm beliefs, tenacious fighting spirit and enterprising spirit of exploration are the best examples of educating students.
How to Infiltrate the History of Mathematics in Middle School Mathematics Teaching
George W. William said: "History has no real scientific value, but its real purpose is to educate others." As a quasi-math teacher, we should not only learn the history of mathematics, but also learn to use it. If teachers purposefully and systematically integrate into the history of mathematics in mathematics class, it will not only enrich the teaching content, but also exert a subtle influence on students and benefit students and doctors. So how to infiltrate the history of mathematics in middle school mathematics teaching? Here are some common methods:
4. 1 Skillful use of famous topics in mathematics history teaching
In the long history of the development of mathematical history, famous topics in the history of mathematics have played an important role in the supplement and development of mathematical knowledge, such as the problem of "chickens and rabbits in the same cage" in Sun Tzu's Calculations, the three major geometric problems in ancient Greece, Goldbach's conjecture and so on. These famous historical questions generally have a certain realistic background, which reveals substantive mathematical methods and is of great help to students in understanding the contents and thinking methods of mathematics.
On the Role of Mathematics History in Middle School Mathematics Teaching Through the presentation of the famous open history questions by teachers, on the one hand, students can understand that the field of mathematics is moving, active and unfinished, and it is not a static and closed system. On the other hand, it can also make students realize that mathematics develops in the process of guessing, making mistakes and repeating mistakes, and the progress of mathematics is an innovation of traditional concepts, thus stimulating students' thinking and making them feel how exciting it will be to master appropriate and valuable mathematical problems.
For example, the proof of Pythagorean theorem, a famous theorem in elementary geometry, has attracted many people because of its simplicity and wide application. Because of the long history, it is difficult to know who is the first person to prove Pythagorean Theorem, but there are various ways to prove it, including Zhao Shuang's proof, American President Garfield's proof, Euclid's proof, similar triangles's proof and so on. Telling students the history of Pythagoras' geographical proof can make the boring proof process interesting and humanized. What is important is to make students feel that they are exploring knowledge and let them participate more actively. So many famous people in history have proved the geography of Pythagoras. Now, with those famous people studying the same problem, this problem becomes different. Even if someone has proved it in the same way in history, when a student solves the proof of Pythagorean theorem alone, his sense of accomplishment and pride can't be compared with other successful acquisitions. This sense of accomplishment will also make students have a strong interest in mathematics from now on.
4.2 Using the history of mathematics to introduce new courses
As the saying goes, "A journey of a thousand miles begins with a single step". A good beginning is half the battle. Teachers can use the history of mathematics to introduce new lessons, attract students' attention, guide students' thinking from the knowledge of the last lesson, and make the classroom reach the best psychological state, thus improving the efficiency of learning. At the beginning of math class, teachers should properly teach students some stories and legends about the generation of math knowledge, which can not only arouse students' direct interest in knowledge points, but also let students see the generation and development process of knowledge. Of course, to do this, the teacher must be carefully designed, strive to be fascinating, take charge of the overall situation, and cause * * *.
For example, when talking about geometric series, the teacher can tell the students the story that the king of ancient India rewarded the wise with wheat: It is said that there was a king in ancient India who liked playing chess very much. One day, a wise man played chess with the king and won the king. The king said he could meet one of his demands. The wise man's request is that the king put 1 grain in 1 grid and put two grains in the second grid. There are four grains in the third box, and so on. The number of grains in the last grid is twice that in the previous grid (there are 64 grids in the chessboard). I hope the king can give him these wheat. The king thought it was not easy, so he readily agreed to his request. After calculation, the total number of wheat grains needed by the inventor is 2 to the 64th power minus 1, which is very large. Introducing this story into the new course of geometry series is believed to attract students' attention and cultivate their interest in learning mathematics. The desire of robot students to explore new knowledge will make students feel high, thus creating a good classroom atmosphere.
4.3 Use the history of mathematics to set the end of the class.
Whether a class is good or not depends on the end of the class. At the end of the class, it is mainly to realize the teaching sublimation of this class, to assist students to summarize and refine the knowledge points, so that they can sort out the overall thinking and master the deep connotation of knowledge in the teaching process. In addition, a good end-of-class link can also play a connecting role, make students interested in the content of the next class and pave the way for the smooth progress of the next class. At this time, if teachers can make good use of the knowledge of the history of mathematics to end this class, they can not only attract students' interest, but also stimulate students' imagination and explore the mysteries of mathematics knowledge. Not only that, because each student's learning level and needs are different, taking the history of mathematics as the end of the classroom can make students with different foundations develop in different degrees, let students with solid foundations continue to explore in depth, and also inspire relatively backward students.
For example, teacher Chen Jingrun told the students at the end of the class "The Nature of Integers": "In natural science, mathematics is the queen, and the crown on the queen's head is number theory. Goldbach conjecture is the most dazzling pearl in this crown. Many mathematicians have devoted their lives to this pearl. I wonder who will pick this pearl in the future? " It was this teacher who remembered the seeds of students' inquiry with the knowledge of the history of mathematics at the end of the class, and only then did the first person in the world conquer Goldbach's conjecture.
4.4 Use the history of mathematics to teach a series of knowledge
Every series of mathematical knowledge is gradually developed through a long historical evolution, and every link of knowledge is acquired at the expense of countless energy and setbacks from generation to generation. Mathematics teaching should achieve the unity of history and logic, and find the right time to let students experience and appreciate the necessity and basic methods of mathematical creation like mathematicians of that year. In the process of mathematics teaching, teachers can regard what students have learned as a link, each link is connected into a knowledge system through historical time and events, and systematically discusses the process and development of knowledge in each link to students. If the teaching progress permits, teachers can conduct appropriate special study and introduce some knowledge of the history of mathematics to students, such as knowledge background, knowledge influence and practical application in real life. Sorting out the mathematical knowledge in students' minds to form a relatively clear and complete system will have the effect of 1+ 1 ~ 2.
Taking the development history of numbers as an example, in production activities, people have produced the concept of natural numbers in order to measure the quantity of commodities, and have produced scores in the division of commodities. In order to represent quantities with opposite meanings, positive and negative numbers have been introduced, irrational numbers have been introduced in the measurement of continuous quantities, imaginary numbers have been introduced from the point that negative numbers cannot be squared, and real numbers have been extended to complex numbers. In this way, the general trend of the theoretical development of numbers is formed: natural number-integer-rational number-irrational number-real number-complex number, which makes students clear at a glance and is very conducive to cultivating students' view that knowledge is changing and developing.
4.5 Use the history of mathematics to carry out inquiry learning
The activities of mathematical knowledge are obtained through observation, experiment, communication, analysis, synthesis, reasoning and summary, but this long and complicated process is rarely reflected in our textbooks. Teachers can take the history of mathematics as the carrier to analyze several key features of a concept. When learning this concept, thinking learners may feel some difficulties. They only understand the superficial meaning of the concept, but do not understand the deep meaning of the concept. However, if we explain the development process of mathematical concepts to students according to their cognitive rules, disassemble and understand mathematical concepts, then serialize and reconstruct knowledge, and then implement teaching on this basis, so that learners can reproduce several key inquiry activities that mathematicians experienced in the process of concept formation under the guidance of teachers, and at the same time, teachers can give them appropriate guidance and let them experience the original thinking process, which will not only enrich students' learning content, but also increase their interest in the history of mathematics.
In inquiry learning, the history of mathematics also plays a very common role, that is, creating inquiry learning situations, which should take into account all factors, be attractive, be authentic, conform to students' real life, and also take into account the regularity and order of knowledge generation and development. Then it is more appropriate to use the history of mathematics to create the scene of inquiry activities, which is not only conducive to the development of inquiry learning but also plays a role in cultivating students' culture. For example, when a teacher teaches the knowledge of "equal possibility events", he can tell students what happened in the field of mathematics today. This series of mathematical events happened on this day. Is this just a coincidence or a normal phenomenon?
5 abstract
To sum up, the history of mathematics is not only to stimulate students' interest in learning mathematics, but also to help students understand mathematical knowledge and master mathematical thinking methods. It plays an important role in cultivating students' exploration spirit without fear of difficulties and obstacles. In mathematics teaching, it is more necessary to use the resources of mathematics history to promote education and teaching. If used well, it can make math class more vivid and infectious. In order to serve the practice, theory can penetrate into the history of mathematics through various methods, including: skillfully using the famous topics of the history of mathematics to teach, introducing new courses, setting the end of classes, teaching knowledge series and carrying out inquiry learning. The above is my personal experience. Due to the limited level, please forgive me if there are any shortcomings.