2. 1 Let students absorb the original juice of knowledge, which will help students better understand and accept knowledge, show mathematicians' thinking, enlighten students' wisdom and cultivate students' creative thinking.
Anyone who has studied mathematics may have had such an experience. When we first came into contact with the concept of "using letters to represent numbers", we used symbols to represent some concepts, such as addition, subtraction, multiplication, division, fractional symbols, decimal notation, logarithmic symbols and limit symbols. There will always be some confusion, and we don't understand why we should express them like this. So everyone can only reluctantly and vaguely accept it. For example, some definitions, theorems and so on. And the teacher didn't show how they were obtained, so everyone has to memorize these things and it is difficult to use them flexibly. In fact, mathematics is both creation and discovery. From science itself to definitions, theorems and mathematical symbols, it is produced and developed from certain thinking under certain cultural and historical background. Lenin said: "The history of a science is the most precious part of this science. Science can only give us knowledge, and history can give us wisdom. " Mathematics teaching without paying attention to wisdom training has no future. As far as cultivating students' mathematical thinking ability is concerned, the experience and lessons in the development of mathematical thinking in the past are the most enlightening, and mathematical knowledge will not be produced out of thin air without the history of mathematics. "Bad teachers reveal the truth, and good teachers teach people to discover the truth." Therefore, our mathematics education should strive to restore and reproduce this development process and look for the source of knowledge from the mathematician's wastebasket. For example, when talking about "negative numbers" in algebra, the teacher should introduce the main reasons for negative numbers: one is the need from life practice, and the other is the need to solve equations. For example, you 12 years old and your father is 35 years old. When will your father be three times your age? Solve equation 35+X=3( 12+X) to get X=35-36. It is found that this result cannot be expressed by a positive number we have learned before, and it must be extended to another number, and the result is negative. Another example is the introduction of the symbol "". Like the introduction of negative numbers, we should first make clear the necessity, rationality and objectivity of introducing the symbol "",and then give it meaning. This kind of teaching can make students understand how some symbols and concepts in mathematics come from, see their functions, realize that the generation of a symbol or a number is a natural and objective demand and a product of human progress, and realize that human genius lies in creation. When a problem seems impossible, people can create some new characters or forms to express a new concept or viewpoint. Students no longer have doubts, but naturally accept this knowledge, which also plays a certain role in cultivating their creative consciousness. If there is an opportunity, teachers can also tell stories about this part of knowledge, such as "negative numbers that have gone through hardships", "the birth of irrational numbers and the first mathematical crisis", so that students can understand how difficult it is to produce these symbols or concepts, and thus sprout the idea of mastering this knowledge. Teachers can also introduce some ancient mathematical symbols and theoretical expressions that have been abandoned but are related to the mathematical symbols that can be used now, and compare them, which will help students understand the advantages of these symbols and expressions used now, thus deepening their impressions and using them better.
When learning some important theorems and definitions, we also encourage teachers to properly show students their own discovery process and mathematicians' thinking process. For example, in the study of Pythagorean Theorem, we should introduce the proof method of Liu Wei's principle of "complement and complement" to students, and show his proof process, which shows that his proof method is more concise, intuitive and ingenious than Zhao Shuang's proof of sum and Li's proof method with "string diagram". In this way, students can not only see the result of rigorous argumentation, that is, the record of success, but also see the process of mathematicians' thinking activities, and feel the keen insight and wit of mathematicians, which is of guiding significance for cultivating students' mathematical thinking. Because the exploratory thinking of students in problem-solving activities is essentially the same as that of mathematicians in research activities. Moreover, from the fact that Liu Wei, Zhao Shuang and others can achieve the purpose of proving Pythagorean theorem through different ways, students can also get enlightenment that there are various ways to prove or solve a math problem. In learning, don't be satisfied with a proof method or a solution, but give full play to your wisdom, constantly tap the potential and seek the best solution.
2.2 Cultivate students' national self-esteem and pride, and encourage them to learn math well.
It is our teachers' bounden duty and obligation to educate students in patriotism. China has made outstanding achievements in mathematics and made great contributions to the development of mathematics in the East and even in the world. In mathematics teaching, if we can carry out some achievements or deeds of mathematicians at home and abroad naturally, appropriately, vividly and interestingly in combination with the teaching content, we can cultivate students' national pride and pride and enhance their thoughts and feelings of loving the socialist motherland. This kind of educational effect can hardly be replaced by its way. For example, when we talk about "equation", we can introduce it to students: in the chapter "Equation" of the book "Nine Chapters of Arithmetic" written in China, the general solution of linear equations is given, and its solution is actually the same as the elimination method used in modern times, but the symbols, terms and calculation tools (using chips to calculate in ancient times) are different. It was not until the end of 17, 500 years later, that the German mathematician Leibniz gave the answer. It shows that the solution of China's linear equations is a brilliant achievement in the history of mathematics and the pride of the Chinese nation. For another example, China was the first country to admit negative numbers. The famous limit judgment in Limit is "The value of one foot, if you take half of it every day, it will be inexhaustible." It has a great influence on the development of mathematics in later generations. Liu Hui invented pi, and got the world's first approximate value =3. 14, and Zu Chongzhi was the first person in the world to calculate pi to seven decimal places, more than 1 100 years earlier than Dutch Antuoni in the 7th century ... modern mathematics also has something to be proud of. China,,, etc. It also has a great influence on mathematics ... All these brilliant achievements are the glory of our Chinese nation, inspiring students to inherit and carry forward the brilliant achievements of mathematics in China and make contributions to the development of mathematics in China.
Of course, while introducing these brilliant achievements to students, we should also point out the shortcomings. For example, due to the long-term rule of feudal forces and the oppression of foreign forces, China's productivity is very backward, and mathematics, as the basic theory and application tool of natural science, has developed slowly for a long time. After liberation, under the leadership of the * * * Production Party of China, the productive forces of new socialist China were completely liberated, the cause of socialist construction was developed, and the mathematical research was improved. Show students the reasons for the slow development of mathematics in China, and let students realize that the feudal and decadent social system is the root of preventing scientific development, and the superior social system provides good conditions for scientific development. This can inspire students to love the Communist Party of China (CPC) and the socialist motherland, cherish today's excellent learning environment and conditions, and study harder. At the same time, students should understand that although China has made considerable progress in mathematical research and education, mathematicians in China have made many important achievements, and students in China have also made gratifying achievements in the Mathematical Olympics. Generally speaking, there are not many world-class research results, and some schools have their own ideal characteristics. There is still a gap between China's mathematics career and the world's advanced level. You will be beaten if you fall behind. Chen Jingrun's "1+2" has occupied a place in the world's national forest for China's mathematics career and won glory for the Chinese people. In this way, students are encouraged to strive for the development of mathematics in China and the rise of China, so that they have a correct learning motivation, that is, in addition to the pursuit of knowledge itself, they should also study for the more brilliant scientific cause of the motherland. Psychologists believe that motivation is the internal reason for people to act to achieve a certain goal, and it is also the internal driving force for people to act. Interest is the best teacher, but without the right motivation, interest will follow the feeling and lose the right direction. It is very important for students to establish good and correct learning motivation for them to learn knowledge well.
Teachers can also combine the patriotic deeds of mathematicians to educate students in patriotism. For example, Chen Shengshen, a world-famous mathematician, gave up favorable material conditions abroad and returned to China to devote himself to science. Another example is that just after liberation, many famous mathematicians living abroad, such as Hua, did not fear the temptation of money, broke through many difficulties, and returned to China one after another, contributing their whole lives to the cause of mathematics in the motherland. Hearing these stories, students will resonate strongly and learn the spiritual accomplishment that will be useful for life from these mathematicians. They will understand that our study today is not for ourselves, but for the prosperity of the great motherland. Strong patriotism can resist the temptation of any money or honor. When our students face similar problems in the future, these scientists' deep-rooted patriotism will enable them to make the right choice, instead of forgetting where their "roots" are for the sake of a comfortable life, or even doing things that are sorry for the motherland, as some international students do now!
2.3 Cultivate students' dialectical materialistic world outlook, gradually form a materialistic world outlook, and strengthen the ability to identify.
The history of mathematics itself is the history of the struggle between materialism and idealism. Engels once pointed out that mathematics is "an auxiliary tool and manifestation of dialectical method". Because of the high abstraction of mathematics, it often hides the materiality that it comes from objective reality. In mathematics teaching, if its materiality is not revealed, students will fall into the confusion of idealistic metaphysics and mistakenly think that mathematics is not from objective reality, but the product of arbitrary thinking mentioned by idealism, which was invented by a few "genius" mathematicians. For example, by telling the emergence and development of concepts such as rational number, irrational number, imaginary number and logarithm, students can realize that these numbers are gradually formed and developed with the needs of human life and production. Geometry is also produced and developed from people's actual needs. Such as: point, line, surface, angle, polygon, circle, sector, bow; Cylinders, cones, platforms, spheres and polyhedrons; Ellipse, hyperbola and parabola; Concepts such as similarity, intersection, verticality, parallelism, area, volume, etc.
For another example, when talking about the problem of ruler and ruler drawing in plane geometry, teachers should explain to students the historical origin of the "three major problems of ruler and ruler drawing", make it clear that this is the scientific truth finally established by countless mathematicians' hard work, list all kinds of "proofs" that have appeared, and reveal the scientific principle of ruler and ruler drawing, so as to enhance students' recognition ability. Otherwise, some students will repeat the detours and mistakes in history and spend a lot of time and energy in an attempt to find a so-called new solution to overturn this historically recognized scientific conclusion.
2.4 Cultivate students' spirit of loving science and pursuing knowledge and their strong will to study hard.
Every mathematician's growing experience is a touching story and a chapter that purifies people's hearts. Their love and persistence in science, hard-working spirit, tenacious perseverance and rigorous style have greatly touched students and are of great benefit to mobilizing their non-intellectual factors. Therefore, teachers should combine teaching materials and tell students more touching stories of mathematicians in class or after class. For example, when talking about irrational numbers, such stories can be interspersed. The discoverer of irrational numbers was Hippocrates of the Pythagorean school in ancient Greece. He dared to ask his most authoritative teacher Pythagoras questions, which gave birth to a great discovery in the history of mathematics-irrational numbers, which violated the Pythagorean school's theory that everything was an integer. But in order to stick to the scientific truth and not bow to the forces, he was finally thrown into the sea and became the first mathematician devoted to mathematical truth in the history of mathematics. But real gold is not afraid of fire, and the "irrational number" has not been thrown into the sea with its owner, but has been handed down in society. This story inspires students: (1) every symbol or concept, theorem, etc. It is really hard-won in our mathematics, so we cherish these scientific achievements more and arouse the belief of learning them well. (2) Dare to ask questions in learning and persevere in knowledge, so as to make continuous progress. (3) Truth cannot be extinguished. Believe in and love science. Another example is the story of three generations (father Zu Chongzhi, son Zu Hao) when talking about the "ancestor principle" in solid geometry, which can make students have a strong thirst for knowledge. Zu's life was bumpy, but he worked hard and made many achievements, among which Zu's principles we are learning today are one of his great achievements. Hua was born in a poor family, unable to enter high school and forced to drop out of school. He taught himself mathematics in the monotonous life of standing in front of the counter. Later, due to typhoid fever, he was disabled for life when he was young. However, with his love and persistence in mathematics, he overcame all kinds of difficulties, worked hard and made continuous progress, and became a self-taught student, and achieved a high position in mathematics. Chen Jingrun exhausted his experience, worked hard, overcame many difficulties and explored the mystery of number theory. In order to prove Goldbach's conjecture, he read countless books and made a mountain of manuscripts, and finally achieved good results. Stories, such as quiet spring rain, moisten students' hearts, edify students' sentiments and play a subtle role, which helps students to learn the enterprising spirit of scientists to climb scientific peaks and cultivate their strong will. Psychologists believe that will is an important psychological factor for success. Strong-willed people will consciously strive to achieve their predetermined goals with practical actions, consciously overcome various difficulties, and make their actions obey the established goals and beliefs. Weak-willed people will March forward regardless of difficulties. Therefore, having good will quality is a necessary condition for students' academic success. By cultivating their strong will, they can overcome difficulties, laziness and negative emotions and study hard.