Mathematical Calculus Thesis Model Article 1: Elementary Calculus and Middle School Mathematics Abstract: As a part of higher mathematics, elementary calculus belongs to college mathematics. Under the background of the new curriculum, several middle school textbooks have been published. It can be seen that the advantages and disadvantages of elementary calculus entering middle school are obvious, and its importance is self-evident, but for many in-service teachers, it is still very strange or not well understood. This is not conducive to teaching in this area. I will make a simple research on the background, function and teaching of elementary calculus entering middle school mathematics.
Keywords: calculus; Background; Function; function
First, the background and necessity of calculus entering high school textbooks.
In the history of mathematics development, since Newton and Leibniz founded calculus, many problems in mathematics have been solved. Calculus has become an indispensable knowledge for us to learn mathematics. Its wide application in economy, physics and other fields also makes it an important tool to solve practical problems in life. But what was the calculus created by Newton and Leibniz? I don't know? Calculus, that is, even they can't tell the theoretical basis of calculus, but they can apply it. This makes many people not understand calculus, let alone let middle school students learn calculus.
Cauchy and Wilstrass established a strict limit theory, which consolidated the foundation of calculus. It is the second generation of calculus, but the concept and reasoning are complicated and circuitous, which is beyond the understanding of high school students. In recent ten years, with the unremitting efforts of a large number of mathematicians, such as Chen, Chen, etc., the third generation of calculus has emerged, which is clearer than the previous two generations and easier for high school students to understand. This laid the foundation for its complete entry into high school textbooks. From the content point of view, the concept and application of definite integral have been added in the calculus part of the new mathematics textbook (calculating the trapezoidal area of curved edge, the volume of rotating body and its application in physics), which may be due to the consideration of the cognitive ability of middle school students. The new textbook of PEP is different from that of Beijing Normal University in this respect. Calculating the volume of a simple rotating body by definite integral appears in the textbook of Beijing Normal University, but the People's Education Edition does not appear.
Judging from the curriculum standard and examination outline (refer to 201kloc-0/college entrance examination outline), the proportion of elementary calculus is getting heavier and heavier. Looking back on previous college entrance examinations, the scores of calculus-related questions are getting higher and higher. But from my personal point of view, the role of elementary calculus in middle school mathematics has not really come into play. In my opinion, it is the clue of students' middle school mathematics and teachers' teaching, the unified method of learning middle school functions, and the link between middle school and university mathematics knowledge!
Second, the role of calculus in middle school mathematics
1. Connect and follow up. Calculus belongs to the category of higher mathematics in universities and is a course offered by universities. Let middle school students learn some calculus knowledge in advance, which will lay a good foundation for studying calculus in universities in the future, and also make the mathematics knowledge from primary school to universities more closely linked in content. There will no longer be many college students who think that college mathematics knowledge has no effect in senior high school mathematics teaching.
2. Solve the role of mathematics related knowledge. High school mathematics function should be ranked first in the whole middle school mathematics content, no matter from the proportion of college entrance examination or its own difficulty. It is always the most difficult for students to learn, and the score rate is relatively low. Many students hate math, but they hate functions. They get dizzy at the mention of functions in mathematics. Because of exam-oriented education, students have to learn functions, and function thought itself is also a clue to high school mathematics learning. The introduction of calculus has found a unified method for students to learn function problems. In high school, the function problems we study are generally based on some basic elementary functions to study the definition, images and properties of functions, and of course there are applications. However, with the deepening of curriculum reform, the problem of function application is gradually weakening. Elementary calculus knowledge is an important tool for learning functions. For example, calculus can find monotonicity and maximum value of functions. The most important thing is that it can draw the image of the function. In fact, almost all the properties of the function can be solved when drawing the image of the function. As long as students learn calculus well, they can master the unified method of learning functions, so that the learning of all elementary functions such as quadratic function, exponential function, logarithmic function and trigonometric function in high school can be unified, which not only saves teaching time but also learns advanced mathematical ideas. To lay a solid foundation for improving students' mathematical literacy. I believe it can also stimulate their interest in learning mathematics. In addition, in high school, elementary calculus can also solve many mathematical problems, such as inequality, tangent of quadratic curve, area of curve trapezoid and so on. Using calculus can not only simplify the problem, but also make the research of the problem more in-depth and comprehensive.
3. Improve the application ability of mathematics in other disciplines. As a natural subject, mathematics itself has been applied to social economy, science and technology and other fields. As middle school mathematics, its promotion to other subjects in middle school is also beyond doubt. Such as physics, chemistry, geography and other disciplines are also inseparable from mathematics. In senior high school, the progress of mathematics teaching often affects the progress of other subjects. If you want to learn the latitude and longitude of the earth in geography, you need to learn the knowledge about spheres and triangles in mathematics first. When calculus enters middle school mathematics, the role of mathematics becomes more important. Especially in physics, the problems such as displacement, instantaneous velocity and acceleration of uniformly accelerated linear motion are easier to understand by using the derivative of calculus. In the new curriculum, the elective course 2-2 of the mathematics textbook of People's Education Press has specially added the section of finding the distance of variable-speed linear motion with definite integral. In addition, calculus, which solves the optimization problems in life, has also entered the middle school textbooks. It can be seen that the introduction of calculus into middle school textbooks plays a vital role in promoting the integration of knowledge between disciplines.
Thirdly, some international textbooks deal with calculus knowledge.
Take middle schools in the Soviet Union as an example. The primary and secondary schools in the Soviet Union have a ten-year system. After the ninth grade (1) (equivalent to the first grade of senior high school in China) is finished, infinite sequence and limit are introduced. Then introduce the function limit and derivative, all before explaining trigonometric function, power function, exponential function and logarithmic function. Then it introduces the application of derivative in approximate calculation, geometry (tangent) and physics (learning speed and acceleration) and the application of derivative in learning function problems (finding function extreme value, maximum value, monotonicity, etc.). ). At the end of the ninth grade and the tenth grade (2), I can talk about trigonometric functions, and I can use derivatives to study the properties of trigonometric functions. Then introduce indefinite integral and definite integral. Then in the chapter of exponential function, logarithmic function and power function, the derivative function of exponential function is introduced, and then the derivative function of logarithmic function is obtained by inverse function. The tenth grade (3) uses the knowledge of calculus to study geometric problems, and deduces the volume formulas of cones and spheres. By integration. The surface area of a sphere is defined as the derivative of the volume V(R) of the sphere to r, so that the formula of the surface area of the sphere can be obtained immediately. It can be seen that the concepts and calculations of derivatives and integrals were first introduced in Soviet textbooks, but not explained at last. In this way, the knowledge of calculus can be applied to the study of functional problems, geometric problems and physical problems.
Of course, there are some examples, for example, the teaching materials for middle schools in Taiwan Province Province, and calculus processing are not very different from the current teaching materials I have read, so I will not introduce them again. Handling calculus by appealing is a common way in European middle school textbooks. Its main advantage is to give full play to the role of calculus in middle school mathematics teaching. Make middle school mathematics knowledge more coherent and easy to understand!
Mathematical Calculus Paper Model Part II: Discussion on the Teaching of Introduction to Calculus Abstract: Calculus is an important basic mathematical course for management majors in colleges and universities, and the key to good calculus is to have a good first class. This paper discusses how to teach the introductory course of calculus well from three aspects.
Keywords: calculus; Origin; Content; way
Calculus is a basic course, and the study of this course directly affects the study of specialized courses in the future, while the introductory course plays a guiding role in the study of this course and has a special position and role in the whole course. Introduction class should include the following parts:
First, the introduction of the origin of calculus
Calculus includes differential and integral. Calculus originated from17th century to deal with scientific problems. Firstly, this paper introduces a problem that Fermat, one of the founders of calculus, studied: Suppose a ball is landing, how to find the speed of the ball when it is five seconds behind? If the motion is uniform, the speed is equal to the distance divided by the time. However, the speed here is uneven. Can non-uniform velocity be approximately regarded as uniform velocity? In what way? This is the problem of differential calculus, and then introduces the area problem studied by the ancient Greeks: calculate parabola y=x2, and coordinate axis x is 0? x？ 1 Can you cut the area into n small areas, and then replace the small areas with small rectangles to get the required area from the areas of n small rectangles? The method used here is the integral problem. Some people have studied differential and integral for a long time. The systematic development of calculus began in17th century, and gradually formed a discipline with complete system and strict logic. It is generally believed that calculus was founded by Newton and Leibniz. The key to the development of this system is to realize that the two processes of differentiation and integration are actually reciprocal.
Introduce the anecdotes of Newton and Leibniz mentioned, such as the debate about the priority of creating calculus. Newton told his friends about the related results of calculus in 1665 ~ 1687, and sent the short article Analysis to Barrow, but during this period, no works about calculus were officially published. When Leibniz visited Paris in 1672 and London in 1673, he corresponded with some people who knew about Newton's work. 1684, Leibniz officially published his calculus works. So some people suspected that Leibniz knew Newton's specific work, and Leibniz was accused of plagiarism. After two people died for a long time, the investigation proved that Newton had done a lot of work before Leibniz, but Leibniz was an independent inventor of calculus.
Secondly, introduce the contents and methods of calculus.
The research object of calculus is function, and limit is the most important reasoning method and the foundation of calculus. There are four kinds of calculus: first, it is known that the distance of an object is a function of time, how to get the speed and acceleration of the object at any time from the distance; Conversely, it is known that the acceleration of an object is a function of time, so how to find the speed and distance? The second is to find the tangent of the curve. The third is to find the maximum and minimum value of the function. The fourth is to find the length of the curve, the area enclosed by the plane curve, the volume enclosed by the surface and the center of gravity of the object.
Third, why do you want to learn advanced mathematics?
Calculus has applications in natural science, economic management, engineering technology, life science and so on, and it is a powerful mathematical tool in various disciplines. Learning calculus well can increase the rigor and accuracy of language, exercise people's rational thinking and feel the art of beauty. Like the golden section, the sum of irrational numbers? Expression:
The introductory course of calculus is the first lesson in the whole teaching. The teaching of introduction class can make students have a quick and general understanding of this course, and a good introduction class can guide students to learn actively.
Mathematical calculus model essay Part III: Introduction to the teaching reform and practice of calculus.
Great changes have taken place in science, technology and society in the 20th century. As one of the basic courses in colleges and universities, advanced mathematics plays an increasingly important role in other fields and disciplines. Calculus teaching, in particular, is a major topic in mathematics education at present.
First, the current situation of calculus teaching reform in China
At present, there are still some main problems in the teaching reform of calculus in mathematical experiments.
First of all, insufficient attention is paid to the cultivation of outstanding talents. In the teaching of calculus, we pay attention to the education of popular talents, but not to the cultivation of some top talents.
Second, over-examination. Paying too much attention to exam-oriented education is becoming more and more obvious in calculus teaching, and it has become a tendency of neglecting ability and emphasizing examination.
Thirdly, students differ greatly and their quality declines. The sharp increase in the number of students has strengthened the differences among students. Faced with this situation, how to plan classes and treat students differently is a problem faced by calculus teaching.
Second, the necessity of calculus curriculum reform
With the deepening of higher mathematics reform, calculus teaching reform has become an important part. The reform of calculus teaching is not groundless, but imperative.
(1) Requirements put forward by high social development
Calculus, as a part of advanced mathematics, plays an important role in promoting scientific and technological civilization, and many mathematical details and achievements are inseparable from calculus. It can be said that calculus plays an important and indispensable role in promoting mathematical thought, social progress and scientific development. It is a great achievement of human thinking, not just advanced mathematics. Moreover, it is necessary for other majors and other majors to understand calculus in different scope and degree. Imagine that if the study of calculus is cancelled, the progress of skills will be just empty talk, society will not develop and wisdom will not be fully explored. Therefore, the reform of calculus teaching is very necessary.
(2) the needs of the development of science and technology
Today's world is an era of rapid technological progress. Without the progress of science and technology, the fierce competition in military, trade and market economy will lag behind others. How to promote scientific development? Calculus plays an important role. It not only provides accurate mathematical ideas for science, but also provides theoretical support for science. It has not only changed the face of mathematics, but also been a tool and method for other disciplines. Calculus is applied to all aspects of natural science. With the development of science and technology, it is imperative to improve the quality of calculus teaching.
(3) the needs of the development of human thinking
There are many important ideas in calculus, such as dialectical thought, constants and variables, isolation and development, static change, finite and infinite, and so on. Straight man? With what? Qu? ,? Local? With what? The whole? This dialectical relationship, in fact. Philosophy is closely related to mathematics, so mathematics, especially calculus, is full of logic and dialectics, and calculus learning. It is not only the study of knowledge and theory, but also the training of thinking. Therefore, the perfection of calculus teaching is conducive to cultivating people's thinking, making people's thinking have a leap and solving problems more effectively.
Third, the content of calculus curriculum reform
According to the revision of the new syllabus, calculus teaching has redesigned the course content, teaching ideas and teaching methods, which is more intuitive and vivid with students as the main body, and has also innovated the teaching methods. The reform of calculus teaching has been comprehensively promoted.
1, the reform of the basic concept of curriculum
The key to the success of calculus teaching reform lies in the change of ideas. In the past, we focused on theory, but now we should pay attention to application, stimulate beginners' interest in learning, master the basic knowledge of calculus as soon as possible, and turn the abstract and difficult calculus theory into a calculus teaching method that students can easily accept and understand. For example, limit is a difficult point in calculus knowledge. The concept, exercises and dialectical thoughts of limit are abstract and difficult to understand for students, which can not stimulate students' interest in learning and make the classroom dull. The revision of calculus syllabus also reflects the renewal of teaching concept, and the difficult knowledge is appropriately reduced in the new calculus teaching. Attach importance to the understanding of the essence of calculus, improve students' interest in learning calculus and learning efficiency through intuition and examples, make students' learning initiative return to themselves, embody people-oriented thinking, attach importance to the cultivation of students' emotional attitude and life value, teach students in accordance with their aptitude, and provide students with better learning conditions and foundations.
2. Teaching content reform
According to the revision of the syllabus of the Standard, calculus teaching is to simplify, add or delete the course content and syllabus at first. The revised teaching content is more concise and scientific than the original syllabus. Like the original 12 hour? Limit? It was largely deleted in the revised outline. The concept of derivative is introduced in the revised syllabus, because derivative is the first concept of calculus and plays a fundamental role in understanding the concept of derivative. Moreover, in the revised textbook, the explanation of derivative is intuitive and applied, and there are many examples to help students deepen their understanding. Therefore, the new curriculum reform of calculus teaching reduces the learning burden of students and the difficulty of concept understanding.
3. Curriculum design reform
The original course is designed from the order of limit, continuity, derivative, derivative application, indefinite integral and definite integral. Derivative and differential? What is the previous chapter for? Limit? Many definitions have been designed, right? Limit? Explains the solution and operation of. The revised syllabus has adjusted the course design, especially the route of calculus explanation, from instantaneous speed, rate of change, derivative and derivative application to definite integral. Calculus courses in humanities and social sciences universities are mostly treated as elective courses, which are very close to life and have strong application, so that non-mathematics majors have a certain basic understanding and interest in mathematics.
4. Innovation of teaching methods
The infiltration and application of (1) mathematical thinking method. There are many ways of thinking in mathematics, which have been effectively used in life. Calculus is an aspect of higher mathematics, and it is scientific to introduce mathematical thinking methods into calculus teaching. Among them, mathematical analysis, also called calculus, is a very important mathematical thought, which appeared in17th century. It not only has a very important position in the17th century, but even today, this thinking method is very helpful to successfully solve the operation of infinite process, that is, limit operation. The application of mathematical ideas has become an innovative project that countries attach great importance to.
(3) Strengthen case analysis and application. Mathematics is a kind of logical reasoning. But it also comes from life and is finally applied to life. Therefore, mathematics teaching cannot be divorced from reality. The revised calculus syllabus obviously pays attention to practical application. Even the simple concepts in the book are always interspersed with some practical pictures. In practice, it is closely combined with the reality of life, rather than castles in the air. For example, exponential function is used to look at the problems of bank deposits and population, while logarithmic function involves radioactivity, decibel and earthquake magnitude. The application of calculus mathematics in solving practical problems in life.
5. Innovation of teaching tools.
The application of modern educational technology, especially multimedia technology, in calculus teaching is of great significance for realizing teaching ideas and improving teaching ideas and methods. Limit? Concepts and theories have always been difficult to overcome in teaching, because they are abstract, so it is inevitable that students will not understand no matter how teachers explain them, and the application of multimedia teaching has solved this problem. Teachers can use intuitive animation to express such as? Infinite approximation? This theory can give students an intuitive and perceptual cognition, and can also use multimedia to design animation with variable parameters, so that students can actively participate in the design and deepen their understanding. For example, to understand the concept of derivative, it is necessary to express the instantaneous speed at a certain point at a certain moment with the help of curves. We can make full use of multimedia technology, draw artistic sketches and design animations, so that students can understand the essence of calculus and the concept of derivatives in animation. It is worth noting that when using multimedia technology, we should follow the laws of the subject itself, penetrate repeatedly, step by step, combine teaching materials and actively guide.
Four. abstract