The acceptance of new knowledge and the cultivation of mathematical ability are mainly carried out in the classroom, so we should pay attention to the learning efficiency in the classroom and seek correct learning methods. In class, you should keep up with the teacher's ideas, actively explore thinking, predict the next steps, and compare your own problem-solving ideas with what the teacher said. In particular, we should do a good job in learning basic knowledge and skills, and review them in time after class, leaving no doubt. First of all, we should recall the knowledge points the teacher said before doing various exercises, and correctly master the reasoning process of various formulas. If we are not clear, we should try our best to recall them instead of turning to the book immediately. In a sense, you should not create a learning way of asking questions if you don't understand. For some problems, because of their unclear thinking, it is difficult to solve them at the moment. Let yourself calm down and analyze the problems carefully and try to solve them by yourself. At every learning stage, we should sort out and summarize, and combine the points, lines and surfaces of knowledge into a knowledge network and bring it into our own knowledge system.
Second, do more questions appropriately and develop good problem-solving habits.
If you want to learn math well, it is inevitable to do more problems, and you should be familiar with the problem-solving ideas of various questions. At the beginning, we should start with the basic problems, take the exercises in the textbook as the standard, lay a good foundation repeatedly, and then find some extracurricular exercises to help broaden our thinking, improve our ability to analyze and solve problems, and master the general rules of solving problems. For some error-prone topics, you can prepare a set of wrong questions, write your own problem-solving ideas and correct problem-solving processes, and compare them to find out your own mistakes so as to correct them in time. We should develop good problem-solving habits at ordinary times. Let your energy be highly concentrated, make your brain excited, think quickly, enter the best state, and use it freely in the exam. Practice has proved that at the critical moment, your problem-solving habit is no different from your usual practice. If you are careless and careless when solving problems, it is often exposed in the big exam, so it is very important to develop good problem-solving habits at ordinary times.
Third, adjust the mentality and treat the exam correctly.
First of all, we should focus on basic knowledge, basic skills and basic methods, because most of the exams are basic topics. For those difficult and comprehensive topics, we should seriously think about them, try our best to sort them out, and then summarize them after finishing the questions. Adjust your mentality, let yourself calm down at any time, think in an orderly way, and overcome impetuous emotions. In particular, we should have confidence in ourselves and always encourage ourselves. No one can beat me except yourself. If you don't beat yourself, no one can beat my pride.
Be prepared before the exam, practice routine questions, spread your own ideas, and avoid improving the speed of solving problems on the premise of ensuring the correct rate before the exam. For some easy basic questions, you should have a 12 grasp and get full marks; For some difficult questions, you should also try to score, learn to score hard in the exam, and make your level normal or even extraordinary.
It can be seen that if you want to learn mathematics well, you must find a suitable learning method, understand the characteristics of mathematics and let yourself enter the vast world of mathematics. 2. The story of a mathematician who has experienced mathematics-Su.
Su Yu 1902 was born in a mountain village in Pingyang County, Zhejiang Province in September. Although the family is poor, his parents scrimp and save, and they have to work hard to pay for his education. When he was in junior high school, he was not interested in mathematics. He thinks mathematics is too simple, and he will understand it as soon as he learns it. It can be measured that a later math class influenced his life.
That was when Su was in the third grade. He was studying in No.60 Middle School in Zhejiang Province. Teacher Yang teaches mathematics. He has just returned from studying in Tokyo. In the first class, Mr. Yang didn't talk about math, but told stories. He said: "In today's world, the law of the jungle, the world powers rely on their ships to build guns and gain benefits, and all want to eat and carve up China. The danger of China's national subjugation and extinction is imminent, so we must revitalize science, develop industry and save the nation. Every student here has a responsibility to' rise and fall in the world'. " He quoted and described the great role of mathematics in the development of modern science and technology. The last sentence of this class is: "In order to save the country and survive, we must revitalize science. Mathematics is the pioneer of science. In order to develop science, we must learn math well. "I don't know how many lessons Sue took in her life, but this lesson will never be forgotten.
Teacher Yang's class deeply touched him and injected new stimulants into his mind. Reading is not only to get rid of personal difficulties, but to save the suffering people in China; Reading is not only to find a way out for individuals, but to seek a new life for the Chinese nation. That night, Sue tossed and turned and stayed up all night. Under the influence of Teacher Yang, Su's interest shifted from literature to mathematics, and since then, she has set the motto "Never forget to save the country when reading, and never forget to save the country when reading". I am fascinated by mathematics. No matter it is the heat of winter or the snowy night in first frost, Sue only knows reading, thinking, solving problems and calculating, and has worked out tens of thousands of math exercises in four years. Now Wenzhou No.1 Middle School (that is, the provincial No.10 Middle School at that time) still treasures a Su's geometry exercise book, which is written with a brush and has fine workmanship. When I graduated from high school, my grades in all subjects were above 90.
/kloc-At the age of 0/7, Su went to Japan to study, and won the first place in Tokyo Technical School, where she studied eagerly. The belief of winning glory for our country drove Su to enter the field of mathematics research earlier. At the same time, he has written more than 30 papers, and made great achievements in differential geometry, and obtained the doctor of science degree in 193 1. Before receiving her doctorate, Su was a lecturer in the Department of Mathematics of Imperial University of Japan. Just as a Japanese university was preparing to hire him as an associate professor with a high salary, Su decided to return to China to teach with his ancestors. After the professor of Zhejiang University returned to Suzhou, his life was very hard. In the face of difficulties, Su's answer is, "Suffering is nothing, I am willing, because I have chosen the right road, which is a patriotic and bright road!" "
This is the patriotism of the older generation of mathematicians. 3. Understanding of mathematics. 1. The object of study in the history of mathematics
The history of mathematics is a science that studies the occurrence, development and laws of mathematical science, which is simply to study the history of mathematics. It not only traces the evolution and development process of mathematics contents, ideas and methods, but also explores various factors affecting this process and the influence of the development of mathematics science on human civilization in history. Therefore, the research object of the history of mathematics not only includes specific mathematical contents, but also involves social sciences and humanities such as history, philosophy, culturology and religion, which is an interdisciplinary subject.
In terms of research materials, archaeological materials, historical archives, historical mathematics original documents, various historical documents, ethnological materials, cultural history materials, interviews with mathematicians, etc. are all important research objects, among which mathematical original documents are the most commonly used and important first-hand research materials. From the research goal, we can study the evolutionary history of mathematical thoughts, methods, theories and concepts; We can study the interactive relationship between mathematical science and human society; You can study the history of the exchange and dissemination of mathematical ideas; You can study the life of mathematicians and so on.
The task of studying the history of mathematics is to find out the basic historical facts in the development of mathematics, reproduce its original features, and make scientific and reasonable explanations, explanations and evaluations on mathematical achievements, theoretical systems and development models through these historical phenomena, so as to explore the laws and cultural essence of the development of mathematical science. As the basic methods and means to study the history of mathematics, there are often historical textual research, mathematical analysis, comparative research and other methods.
Historians' duty is to tell history according to historical materials, and seeking truth from facts is the basic principle of historiography. Textual research has been formed in western history since17th century. It appeared earlier in China, especially in the Ganjia period of the Qing Dynasty, and it is still the main method of historical research. However, with the progress of the times, textual research methods are constantly improving and its application scope is expanding. Of course, it is necessary to realize that the historical materials are true or false, and the psychological state of the researchers involved in the research process will inevitably affect the choice of research materials and the results of the research. In other words, the authenticity of the conclusions of historical textual research is relative. At the same time, we should realize that textual research is not the ultimate goal of historical research, and the study of mathematical history cannot be textual research for textual research's sake.
If you can't compare, you can't think. All scientific thinking and investigation are inseparable from comparison, or comparison is the beginning of understanding. The development of today's world is multipolar, and different countries, regions and nationalities develop together in cultural exchanges. Therefore, with the development of the research on the history of multi-world civilization and the weakening of western centralism, heterogeneous regional civilization has been paid more and more attention, and the comparison of mathematical cultures in different regions and the research on the history of mathematical communication have become increasingly active. The comparative study of the history of mathematics often revolves around three aspects: mathematical achievements, mathematical scientific paradigm and the social background of mathematical development.
The history of mathematics belongs to both the field of historiography and the field of mathematical science. Therefore, the study of the history of mathematics should follow the laws of both history and mathematics science. According to this feature, mathematical analysis can be used as a special auxiliary means in the study of mathematical history. In the absence or lack of historical data, we can analyze the contents and methods of ancient mathematics from the height of modern mathematics, so as to trace the source, summarize the theory and put forward historical hypothesis. Mathematical analysis is actually a connection between "ancient" and "present".
Second, the stage of the history of mathematics
The development of mathematics has stages, so researchers divide the history of mathematics into several periods according to certain principles. At present, the academic community usually divides the development of mathematics into the following five periods:
1. The embryonic period of mathematics (before 600 BC);
2. The period of elementary mathematics (from 600 BC to1mid-7th century);
3. The period of variable mathematics (1mid-7th century to11920s);
4. Modern Mathematics Period (65438+11920s to World War II);
5. The period of modern mathematics (from the 1940s).
Third, the significance of the history of mathematics
(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 our 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) the cultural significance of the history of mathematics
M Klein, an American mathematical historian, 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) the 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 in the department of mathematics 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, which will 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 people, mathematics is a boring subject, so many people regard it as a daunting task. 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 educational situation, the gap between arts and sciences leads to the fact that the talents trained by our education are 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. 4. Teach students how to learn math well
Junior high school students must solve two problems to learn mathematics well: first, understanding; The second is the method.
Some students think that learning to teach well is to cope with the senior high school entrance examination, because mathematics accounts for a large proportion; Some students think that learning mathematics well is to lay a good foundation for further study of related majors. These understandings are reasonable, but not comprehensive enough. In fact, the more important purpose of learning and teaching is to accept the influence of mathematical thought and spirit and improve their own thinking quality and scientific literacy. If so, they will benefit for life. A leader once told me that the work report drafted by his liberal arts secretary was not satisfactory, because it was flashy and lacked logic, so he had to write it himself. It can be seen that even if you are engaged in secretarial work in the future, you must have strong scientific thinking ability, and learning mathematics is the best thinking gymnastics. Some senior one students feel that they have just graduated from junior high school, and there are still three years before their next graduation. They can breathe a sigh of relief first, and it is not too late to wait until they are in senior two and senior three. They even regard it as a "successful" experience to "relax first and then tighten" in primary and junior high schools. As we all know, first of all, at present, the teaching arrangement of senior high school mathematics is to finish three years' courses in two years, and the senior three is engaged in general review, so the teaching progress is very tight; Second, the most important and difficult content of high school mathematics (such as function and algebra) is in Grade One. Once these contents are not learned well, it will be difficult for the whole high school mathematics to learn well. Therefore, we must pay close attention to it at the beginning, even if we are slightly relaxed subconsciously, it will weaken our learning perseverance and affect the learning effect.
As for the emphasis on learning methods, each student can choose a suitable learning method according to his own foundation, study habits and intellectual characteristics. Here, I mainly put forward some points according to the characteristics of the textbook for your reference.
L, pay attention to the understanding of mathematical concepts. The biggest difference between high school mathematics and junior high school mathematics is that there are many concepts and abstractions, and the "taste" of learning is very different from the past. The method of solving problems usually comes from the concept itself. When learning a concept, it is not enough to know its literal meaning, but also to understand its hidden deep meaning and master various equivalent expressions. For example, why the images of functions y=f(x) and y=f- 1(x) are symmetrical about the straight line y = x, but the images of y=f(x) and x=f- 1(y) are the same; Another example is why when f (x-l) = f (1-x), the image of function y=f(x) is symmetrical about y axis, while the images of y = f (x-l) and y = f (1-x) are symmetrical about the straight line x = 1.
2' Learning solid geometry requires good spatial imagination, and there are two ways to cultivate spatial imagination: one is to draw pictures frequently; Second, the self-made model is helpful for imagination. For example, the model with four right-angled triangular pyramids is much more seen and thought than the exercises. But in the end, it is necessary to reach the realm that can be imagined without relying on the model.
3. When learning analytic geometry, don't treat it as algebra, just don't draw it. The correct way is to calculate while drawing, and try to calculate in drawing.
On the basis of personal study, it is also a good learning method to invite several students of the same level to discuss together, which can often solve problems more thoroughly and benefit everyone.
I hope I can give you some inspiration. Thank you for your adoption.