Understanding of Mathematics and Applied Mathematics Paper 1
The new curriculum reform pays attention to the occurrence and development of knowledge, trains students to observe the society and think about problems from the viewpoint of mathematics, strengthens the consciousness of applied mathematics, and attaches importance to the combination of practice and applied mathematics consciousness. Teachers should strengthen mathematics application teaching, let students learn independently, attach importance to extracurricular practice, promote students to gradually form and develop mathematics application consciousness and improve their practical application ability.
Keywords: mathematics application, life experience, applying what you have learned.
The new curriculum reform pays attention to the occurrence and development of knowledge, trains students to observe the society and think about problems from a mathematical point of view, enhances the consciousness of applied mathematics, and truly makes students realize? Apply what you have learned? . In recent years, I adhere to the new curriculum standards as the guiding ideology, attach importance to practice, strengthen the cultivation of students' mathematical application ability, and do some exploration. Here I want to talk about some thoughts on this issue.
First, the theoretical basis
1. The development of mathematics is the history of its application.
Judging from the early development of mathematics, mathematics originated from the needs of human real life. In the simple commodity exchange and redistribution, human beings have produced the concept of number. There are many geometric problems in the earliest mathematical works "Rhine papyrus" and "Moscow papyrus" handed down from ancient Egypt, most of which are related to the calculation of land area and grain pile volume; China's earliest extant mathematical work "Zhou Bi suan Jing", the main achievement is Pythagorean theorem and its application in astronomical measurement.
In modern times, especially in modern times, on the one hand, the core research of mathematics has become more and more abstract; On the other hand, the application of mathematics is becoming more and more extensive. Mathematics is not only widely used in physics, chemistry, biology and other natural sciences, but also widely used in economics and sociology. In the increasingly developing information society, even ordinary workers must have basic mathematical operation ability, and the ability to observe and analyze their work, life and even engage in economic and political activities by using mathematical ideas-deposit, interest, stock, investment, insurance, cost, profit, discount, installment payment, etc. It can be said that mathematics is one of the most basic and important tools for human activities.
2. The new curriculum reform strengthens the embodiment of mathematics application.
The new curriculum standard emphasizes that starting from students' existing life experience, students can personally experience the process of abstracting practical problems into mathematical models and explaining and applying them, so that students can gain an understanding of mathematics and make progress and development in thinking ability, emotional attitude and values. The new curriculum standard emphasizes the cultivation of mathematics application consciousness, which makes students realize that there is a lot of mathematics information in real life and mathematics has a wide range of applications in the real world; In the face of practical problems, we can actively try to use the knowledge and methods we have learned from the perspective of mathematics to find strategies to solve problems; When faced with new mathematical knowledge, we can actively look for its actual background and explore its application value.
The new curriculum standard puts forward that the content of mathematics learning should be realistic and meaningful. Since the implementation of the new curriculum reform, the newly compiled textbooks have made many improvements in strengthening the awareness of applied mathematics, and the cultivation of students' awareness of applied mathematics runs through the compilation of textbooks. In the chapters, charts or reading materials of each chapter, attention should be paid to providing questions with practical background. Generally, the text of the textbook should pay attention to introducing concepts from reality, asking questions from reality, and adding practical application content to examples and exercises. The purpose of integrating theory with practice is to better grasp the basic knowledge, increase the awareness of applied mathematics, and cultivate the ability to analyze and solve problems. For example, education savings are related to savings in economic life, and topics related to quadratic function, such as braking distance and quadratic function, data collection and processing, statistics and probability, contain a lot of contents closely related to practical problems. The new textbook also adds subject learning, with the purpose of applying the learned mathematical knowledge, improving the ability to solve practical problems, and allowing students to exercise and improve in the process of participating in mathematical activities.
Therefore, as a mathematics teacher, we should pay attention to strengthening the teaching of mathematics application in teaching activities, promote students to gradually form and develop their awareness of mathematics application, improve their practical ability, and cultivate qualified applied talents for the society.
Second, teaching practice
1. Strengthen intuitive teaching and cultivate students' application consciousness.
The introduction of some mathematical problems should provide students with rich and typical perceptual materials by intuitive means according to the teaching content, such as using objects, models, wall charts, or giving demonstrations to guide students to observe and let them operate by themselves, so as to enrich their perceptual knowledge. On the basis of teacher's vivid description, it is particularly necessary for students to understand the value and background of what they have learned in teaching. Students should see when and how mathematics is applied, instead of being promised that it will be used one day. Teachers should emphasize the application of mathematics to problems in the real world and other environments related to middle school students when raising and studying problems.
For example, talking about? Solve right triangle? At the same time, we can use a practical problem: when building a pumping station, we should set up auxiliary water pipes along the slope. What is the slope and horizontal plane from the sectional view? A can be measured by goniometer, and the length of water pipe AB can also be measured directly. When the water pipe is laid at B, let the horizontal plane from B be BC. If you are a construction worker, how do you measure the height of B from a horizontal plane? Some students proposed to drill a hole from B to C and measure the depth of the hole. Some students object, because it is laborious according to the actual situation; Some students also said that this is not a laborious problem, and point C cannot be determined. When teaching, we should pay attention to abstract the mathematical model from practical problems and solve it with the knowledge of solving right triangle: BC=AB.sinA(AB, one is known). Another example is to find the lowest cost of the pool with the knowledge of inequality, to calculate the duration of typhoon influence with trigonometric function, to analyze the secret of free lottery with probability knowledge and so on. Through the application of mathematics in other sciences and social life, let students know that it is not only related to almost all human activities, but also beneficial to everyone who is really interested. Only in this way can the enthusiasm of students be fully mobilized.
2. Set aside time to enhance students' awareness of independent application.
For most students, the way they learn mathematics is still accustomed to listening to the teacher in class, thinking that the more they listen to the teacher, the more they know. Although students have some insights in their study, their basic knowledge is not solid, and their acceptance of a large amount of knowledge and information is relatively rigid, but their understanding is not deep and their flexible application is not in place. In this way, once students leave the teacher, they can't meet some expanding or research problems, so most of them give up. As a teacher, we should set aside more time for students, strengthen guidance and let students stay in school. Autonomous? Learning and being? Cooperation? Strengthen the application of knowledge in exploration and let the application of mathematics be implemented.
For example, when I was reviewing axisymmetric knowledge, I asked a question: There is a village A and a warehouse B on the same side of a river L. One day, the warehouse suddenly caught fire, and the villagers set out from home and went to the river with buckets to put out the fire. So what route should they choose? Because I have done similar exercises before, the students quickly gave the answer: Make a symmetrical point A about Xiaohe L? , and then link a? When B intersects L at point P, the dotted line APB is the route taken by the villagers. I asked the students:? Do you all think so? The students answered in unison. Yes! ? I didn't say anything. I'm just saying: you can communicate again. ? At first, there was a lot of noise in the classroom. What is there to discuss? Isn't it APB Slowly, the voice in the classroom became lower and the students began to think and communicate. Later, the voice in the classroom grew louder. Then I asked: Do the students have any new ideas? A dozen students raised their hands. I asked one of them to speak, and she said, after our discussion, we found a more suitable route. Considering that the bucket full of water is relatively heavy and inconvenient to carry, it is necessary to shorten the distance to carry water. What we do is let BQ? L, the vertical foot is Q, and connecting AQ and broken line AQB is a more suitable route. ? I said:? Do the students agree with her? Most students agreed. After discussing such problems, students have strengthened their awareness of practical application.
3. Strengthen extracurricular application practice.
Practice plays an important role in understanding, mastering and skillfully applying knowledge. What you hear will be forgotten, what you see will be remembered, and what you have experienced will be understood and used. Therefore, extracurricular practical activities should be strengthened. For example,? What is the shortest vertical line segment? After learning nature, let students take advantage of physical activity time to make long jump and measure their long jump performance; After learning the basic knowledge of statistics, let students estimate the fluctuation of their academic performance, and so on. In this way, students not only understand the knowledge, but also learn how to solve practical problems. Often let students practice and use what they have learned to solve practical problems, and students' awareness of applying mathematics will gradually form, which is also an effective way to change educational concepts and implement quality education in classroom teaching.
For example, after data collection and processing, students are assigned to choose appropriate topics, independently design survey plans, carry out survey activities, process data and write survey results. During this period, teachers play an organizational role, do not do specific work, but give appropriate help and guidance to students when they need it, thus stimulating students to actively carry out inquiry activities. On the basis of students' personal experience of the whole process of investigation activities, we should raise awareness again, strengthen statistical awareness and concepts, solve related problems by using statistical methods, and cultivate students' application awareness and practical ability in the activities.
In a word, mathematical knowledge comes from life. In mathematics teaching, teachers should pay attention to students' learning activities, fully tap the mathematical materials in life, cultivate students' habit of observing and analyzing things around them with mathematical views, and solve problems with mathematical methods.
References:
Li Wenlin. History of mathematics development.
[2] Robert. Teaching Thinking, translated by Zhang, et al. China Light Industry Press, 2008+0.
Understanding of Mathematics (Ⅱ) —— Educational Value of Mathematics Culture
Mathematics is an important part of human culture and plays an important role in human civilization and social progress. The educational value of mathematical culture lies in its unique contribution to human rational thinking and creative thinking. Every modern person needs to receive mathematics education, improve his cultural quality through his knowledge and understanding of mathematics, and thus create a more meaningful and meaningful human culture.
[Keywords:] rational creativity of mathematical culture education
Mathematics has three principles of general culture, namely, relevance, compatibility and universality. Relevance is mainly related to reality, not illusory things suspended in mid-air; Compatibility not only emphasizes that it is a logically closed system, but also embodies a multicultural activity mode; The popularity reflects the openness to everyone who studies and practices. In addition, the more important aspect is the particularity of mathematics relative to the general popular culture, which constitutes the personality of mathematics culture, that is, the unique language system, value judgment standard and development model, which makes mathematics itself constitute an independent cultural system, thus making the artificiality of mathematical objects, the integrity of mathematical activities and the historicity of mathematical development full of humanistic values and highlighting the cultural significance of mathematics.
Mathematics and Ancient Culture
Mathematics between China and the West, in the long ancient times, can be essentially attributed to the mathematics of Greece and China, so our comparison is limited to the mathematics and culture of Greece and China.
One of the characteristics of ancient Greek culture is advocating rationality-in mathematics, advocating deductive reasoning and closely linking mathematics with philosophy. Ancient Greek mathematicians emphasized strict reasoning and the conclusions drawn from it. What they care about is not the practicality of these achievements, but educating people to make abstract reasoning and inspiring people to pursue ideals and beauty. Pythagoras proposed it? Graphics and beliefs? Explain the life belief from geometry learning to a higher level, that is, mathematics education and mathematics learning can not take the attitude of quick success and instant benefit. Therefore, the beautiful literature of ancient Greece, extremely rational philosophy, idealized architecture and sculpture, all these achievements have an important position in human history, and these achievements reflect the influence of mathematics everywhere.
Points, lines, surfaces and numbers in ancient Greek mathematics are all idealizations and abstractions of reality, and this preference for idealization and abstraction of reality has also left a deep imprint on its culture. Their sculptures do not pay attention to individual men and women, but to ideal models. This pursuit of idealization and abstraction leads to the pursuit of standardization of the proportion of all parts of the body. The Greeks not only gave the standard golden section of 0.6 18, but also did not ignore any ratio of fingers and toes. Greek culture is recognized as a brilliant page in human history, which has a far-reaching impact on the development of human culture.
China's ancient mathematics paid more attention to practicality and demanded solving problems. In modern terms, it just pays more attention to it? Constructive? Mathematics, not the pursuit of structural perfection and theoretical integrity. This expression is similar to that of China's ancient philosophy. Feng Youlan pointed out in A Brief History of China:? Philosophers in China are used to expressing their thoughts in the form of famous sayings, metaphors and examples. The whole book of Laozi is full of famous sayings and fables, and most famous works of Zhuangzi are full of metaphorical examples. ? These are enough to show the close relationship between China mathematics and China culture.
In ancient China, numbers were endowed with ethical significance. Etiquette is often called? Polite. Because there are specific figures? Rites are regarded as ethical precepts, such as The Book of Rites? Is it in the ritual vessel? The hall of the emperor is nine feet, the princes are seven feet, the doctors are five feet, and the scholars are three feet? Rules, and then what? Ethics? It is regarded as a social law. Thus, it appears in China culture? How many days? This word? How many days? Represents an irresistible fate.
? Is politeness considered in China culture? Rules? What's the matter? No rules, no Fiona Fang? . China people use mathematical laws (with? Rules? Draw a circle with? Instantly? Draw a straight line. ) to describe and describe the political and social operation, some features of China's traditional mathematics have been integrated into the culture. The biggest influence of mathematics on China traditional culture is a set of worship system about numbers. Today, this system is still deeply rooted in people's daily life.
There is no doubt that mathematics is an important part of human culture. As Steen, special editor of American Science magazine, said:? Mathematics is as important as language, art or religion in human characteristics and history. ? The development and achievements of mathematics have an important influence on the culture to which it belongs. On the contrary, mathematics has different cultural values and characteristics in different cultures.
Mathematics Education and the Cultivation of Cultural Quality
China's traditional mathematics is utilitarian in nature, just as? Six arts? First, it is impossible to accumulate into a rational structure of China culture, which does not have a high position in the corresponding cultural system. Looking for the source, this research for us? Examination culture? China's mathematics education under this background may be useful for reference.
At present, mathematics education in China is often aimed at enabling students to pass the exam with high scores, so as to evaluate teachers' teaching level. Can this short-term and utilitarian educational concept cultivate thinking? Once students don't need exams, the role of mathematics is dead to them. How significant is this kind of mathematics education to the cultivation of people's quality? In my opinion, the key to a person's potential lies in whether he can handle tomorrow's problems well. Mathematics education should be an indispensable cornerstone of an educated person's personal cultural heritage, accompanying him all his life, just as learning language better is for expression, learning art better is for appreciation, and learning mathematics should make him think and distinguish more rationally.
1. the cultivation of rational thinking
As a special form of human rational thinking, mathematics has the following basic characteristics: logic; Abstract; An accurate grasp of the main and basic attributes of things.
The logical form of mathematics refers to very strict thinking in mathematics, which is closely linked from conditions (causes) to conclusions (results), and the causal relationship is very clear. This way of thinking is very important to anyone. For example, to achieve an important goal (why to achieve this goal), the specific implementation plan (how to achieve this goal), what conditions (creation) are needed, what problems (potential) exist, where the most important risks come from, and what means to prevent or resolve risks are all very similar to geometric logic. This characteristic of mathematical thinking is very important for cultivating people's quality, but the ability to be good at reasoning is not innate. Only through education can people develop this potential.
Abstraction is not a unique feature of mathematics, but it is the most typical. Mathematical abstraction abandons all other aspects of things, leaving only a certain relationship or structure. When we use quantitative methods to analyze physical phenomena, chemical phenomena, biological phenomena and social phenomena to reveal the relationship between things, we will find that some seemingly unrelated substances, unrelated things and unrelated people are actually interrelated. For example, the normal distribution in probability theory and mathematical statistics shows that the errors of various random events do not appear randomly, but always follow certain statistical laws.
For example, in an ordinary exam, if the test scores are not normally distributed, it can be considered that there is an abnormal phenomenon in a certain link (such as teaching quality, test paper difficulty, grading standard and examination room discipline). And then what? General exam? It can be broadly called linear algebra, English, business management and so on. For another example, people find that people's various mental or physiological characteristics follow the normal distribution. This provides a theoretical basis for human cultural scholars to study the quality and temperament of different human nationalities, and also provides important parameters for medicine and pharmacology.
Finding out the main attributes of a problem in mathematics is to be good at grasping the most essential content of the problem, which shows that people should grasp the fundamental problem when dealing with the problem. Larry, President and CEO of Honeywell International? Bossidy said? There is no so-called complex strategy in the world, only a complex understanding of a strategy. A strategic report of a business department. If you can't describe your strategy in simple and clear language within 20 minutes, you actually have no strategic planning. ? If we are good at grasping the root of problems and simplifying complex problems, that is the embodiment of wisdom. Then, a work report, in the hands of a person who has been trained in mathematics, will at least eliminate some nonsense and cliches that have nothing to do with the conclusion.
Mathematics has made a special contribution to the development of human rational thinking. Mathematics education in ancient Greece advocated that mathematics was the training of rationality and thinking ability. It is believed that arithmetic is to know the essence of numbers, not to do business in pursuit of truth; Geometry is to train thinking and philosophers. They only regard the practical purpose as a trivial aspect of mathematics education, and the cultivation of rationality is the fundamental purpose of mathematics education. It is relying on this kind of education that reason opens the road of human civilization.
The revival of modern western civilization is essentially the revival of mathematical spirit. Europeans in the Renaissance and beyond not only learned and mastered the achievements of the ancient Greeks, but more importantly, learned the reasoning ability of human beings from them. Europeans inherit the idea that nature has mathematical design, and think that reason can be applied to all kinds of human activities. It was after Tessa in western Europe mastered the spirit of reason and mathematics that modern western civilization was born.
In modern society? The tendency to give up rational thinking is a sign of mass and political instability? . In the process of building a harmonious society between man and nature, we should have rational thinking at all times, and the most effective way to cultivate rational thinking is mathematics education. ? Strengthening mathematics education in higher education to make people understand mathematics, attach importance to mathematics and use mathematics correctly is of strategic significance for developing intelligence and improving our nation's scientific and technological level and thinking ability. ?
To sum up, rational thinking is a historical, scientific and philosophical thinking, a critical thinking, a thinking of seeking common ground while reserving differences, and a higher level of moral reasoning. Through the cultivation of mathematical rational thinking, it is helpful for students not to follow blindly, to be methodical, to be good at thinking, and to establish the will to be unyielding and unyielding.
2. The cultivation of creative thinking
Because of the rigor of mathematics, few people doubt the correctness of mathematical conclusions, which often become a model of truth. In fact, the truth of mathematical conclusions is relative. Even a simple formula like 1+ 1=2 has its shortcomings. For example, in Boolean algebra, 1+ 1=0. Boolean algebra is widely used in electronic circuits.
As the saying goes: learning is expensive and doubtful. Doubt is a critical spirit and a prerequisite for innovation.
In the process of teaching linear algebra, I emphasized that matrix is a numerical table rather than a number when explaining the concept of matrix, but I broke through this thinking framework in block matrix operation.
The above calculation process is complicated, but from the calculation point of view, it greatly improves the operation efficiency of higher-order matrix product and has practical application value. In general, people are always used to the conventional way of thinking, because it can make us think about similar or similar problems, save many steps of exploration and exploration, and avoid detours, thus shortening the time of thinking, reducing the consumption of energy, and seemingly improving the quality and success rate of thinking. As a psychologist said:? People who can only use hammers always treat all problems as nails. ?
However, this mindset often plays a role of hindrance and restraint, which makes people fall into the invisible framework of the old thinking mode and it is difficult to make new explorations and attempts. Routine is the general thinking of people to solve problems. It can skillfully complete some work and solve some common problems by experience, but it is a fool to always look at things with a fixed mindset. Of course, it takes great courage to change and innovate. Some people don't have the courage to change even if they realize the necessity of change. Because once the change fails, he will be greatly hurt. But he didn't see the other side of the problem: if he didn't make changes, he would suffer huge losses in the future, and the changes were successful, and successful changes would open up a whole new field for his career.
In the process of teaching advanced mathematics, I asked the students: I walked to the door of the classroom, half the distance at a time, and asked if I could walk to the door. Answer 1: Don't say it's not a problem to walk out the door. Answer 2: Due to conditions? Walk half the distance at a time? So the distance between people and the gate will always exist, so you will never get to the gate. Answer 3: You can go. Because the distance between people and the gate can be shortened as small as possible, and it can be infinitely small. The third answer is correct. This problem embodies the core idea of higher mathematics-limit. It challenges the brain and stimulates people's imagination. The limit is unfamiliar and familiar, which seems to be beyond our understanding ability, natural and easy to understand. In the process of conquering it, we need to mobilize people's reasoning ability, poetic imagination, creativity and curiosity.
Similar to the above problems, after several years, the final problem itself may not be important to most students, but the application of imagination and long-term thinking in the process of mathematical creation can make them break away from convention, learn to be flexible, find another way, and subconsciously accumulate the impulse to create inventions, so as to face difficulties calmly and face the future happily.
As one of the most effective tools to train people's thinking, mathematics education is no longer appropriate in cultivating organizational ability, sensitivity, intuition and insight. The basic goal of mathematics education is to promote the general development of intelligence regardless of students' future career choices. The ultimate goal of mathematics education is not simply to provide students with tools to solve some specific problems, nor to pave the way for existing professional courses, but to cultivate students' pursuit of rationality (truth) and cultivate a spirit, a down-to-earth and fearless exploration spirit.
Mathematics directly or indirectly affects the thinking of every literate person, promotes people's ideological emancipation, and improves the level of human material civilization and spiritual civilization. It can be said that a culture without advanced mathematics is doomed to decline, and a nation without mathematics as a culture is also doomed to decline.
References:
Sun Xiaoli. Math? Science? Philosophy [M]. Beijing: Guangming Daily Press, 1988.
[2][ America] larry Bossidy. Execute [M]. Beijing: Machinery Industry Press, 2005.
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