High school math textbook _ math textbook

Famous physicist and mathematician Carl? Friedrich? Gauss said: "Mathematics, the queen of science; Number theory, the queen of mathematics. " Hua, a famous mathematician in China, pointed out that "the universe is big, the particles are small, the speed of rockets, the change of the earth, the mystery of biology and the complexity of daily use", and no one can leave mathematics. From aviation to family, from universe to atom, from large-scale projects to business management, all benefit from mathematical science and technology. Mathematics gives people not only knowledge, but also ability, including observing experiments, collecting information, inductive analogy, intuitive judgment, logical reasoning, modeling and accurate calculation. The cultivation of these abilities will benefit people for life. The future world is a modern scientific world, and future science is a mathematical science. Mathematics is the key to science. In this issue, several experts, students and editors' comments on related textbooks will help you better understand this "key". ...

Thoughts and Practice on the Reform of Higher Mathematics Curriculum Content System

Comment on Advanced Mathematics (Second Edition)

Jong Li

The predecessor of Advanced Mathematics (Second Edition) is the Concise Course of Advanced Mathematics (all three volumes) published by 1998. 1autumn of 1996 to1spring of 1998, we made some attempts in the course content system during the teaching of advanced mathematics in the departments of physics, radio electronics and technical physics in Peking University, and published a concise course of advanced mathematics (all three volumes), and won the first prize of excellent textbooks in national universities in 2002. In 2004, the book was completely revised for the first time. As the textbook of the tenth five-year plan, the content was changed from three volumes to two volumes, and it was renamed "Advanced Mathematics".

The main readers of this book are college students majoring in physics. Advanced mathematics (or simply calculus) is of self-evident significance to these majors. However, this course is often difficult and even daunting for some students. The main guiding ideology of this book is to adjust some traditional teaching methods, so that the teaching of calculus can return to primitive, natural, interesting and useful.

In this revision, we have deleted the proofs of some theorems, including the boundedness theorem of continuous functions on closed intervals, the median theorem, the maximum and minimum theorems, and the existence theorem of implicit functions. This deletion does not mean a change in the basic requirements of teaching, on the contrary. The proofs of these theorems appear in the original book, or in the form of appendices, or clearly indicate that they are beyond the requirements of the syllabus and do not need to be taught in class. Nevertheless, writing them in books may mislead teachers or students, blur the basic requirements of teaching and increase the unnecessary psychological burden of teachers and students. It is best to simply delete them. Therefore, this is to make this book more clearly reflect the basic requirements of teaching.

This set of teaching materials also has the following points in content processing for peer discussion:

1. Compared with the traditional textbook, this textbook has made some adjustments in the order of teaching content.

At present, most advanced mathematics textbooks in China are calculus first and then calculus. The advantage of this is that the mathematical theory system is clear, but the disadvantage is that the concept of integral comes out too late, which makes beginners feel out of touch with the concepts of differential and integral. In addition, due to the late appearance of the concept of integral, mathematics is seriously out of touch with other courses, such as mechanics and general physics.

In this textbook, we first put the basic concepts of calculus and calculation together, and then talk about the differential mean value theorem and Taylor formula after we finish the basic theorem of calculus and the calculation of integral. The main purpose of this adjustment is to let beginners understand and master the basic idea of calculus as soon as possible and master its core, most useful and most vivid part. At the same time, this adjustment also eased the contradiction with other courses. Therefore, we think that this adjustment may be one of the ways to solve the disconnection between physics major and other basic courses in freshmen.

Calculus is simple and natural in terms of its original core idea and form, and it is easy to be understood and accepted. With the development of history, the strengthening of logical foundation and the deepening of various studies, it has become a "behemoth", which makes beginners daunting. Now, how to choose important things and how to show them to students in a short time can't be said to be a problem, which deserves our consideration and exploration.

Second, the handling of the concept of limit.

The concept of limit and the treatment of real number theory have always been one of the focuses of debate in the teaching reform of calculus. We believe that the strict definition of limit, namely "ε-δ" and "ε-N", should be seriously discussed, because we have to deal with some complicated limit processes, especially when it comes to the uniform convergence of function series. Students majoring in physics may have to learn a lot of advanced mathematics, and it is impossible to master the strict definition of limit.

However, we don't agree to spend a lot of effort to train "ε-δ" repeatedly from the beginning, thus forming a "big head limit theory". We hope that with the deepening of the course, students will gradually become familiar with and master it in repeated use. In the current textbooks, there are not many exercises to prove the limit of a specific function with ε-δ, let alone extreme exercises with great difficulty, because it is of little significance to do too many such exercises. The concept of limit is rigorous in this set of textbooks, while retaining its simple, intuitive and natural character.

Closely related to the concept of limit are several theorems about the completeness of real number field. We adopted a decentralized approach. At the beginning of the book, finite monotone bounded sequences are introduced as mathematical descriptions of the completeness of real numbers. With it, it is enough in many discussions about limits. The first chapter only describes the properties of continuous functions on closed intervals, and there is no proof. The proof is only used as an appendix for interested readers to read. Before discussing series, the completeness of real number field is involved again, and then Cauchy convergence principle is introduced to meet the needs of series discussion. This decentralized method not only disperses the difficulties, but also makes it easier for beginners to see the significance of these basic theorems in the problems involved.

Third, this book adheres to the basic content and basic training in traditional textbooks, but broadens the scope of content.

In the choice of content, we have adopted a very cautious attitude. In recent years, the call for modernization reform of higher mathematics curriculum content is very high. However, as a basic mathematics course, it seems inappropriate to simply take modernization as the main goal of its reform. The consistency of concepts in mathematics makes it impossible to "update" and "bring forth new ideas" like electronic equipment. For a science and engineering student, it still seems essential to master the basic concepts, theories and training of calculus. Of course, the extensive use of computers and the increasingly perfect functions of mathematics software are prompting us to think about simplifying or reducing some calculation contents in higher mathematics classes. But as far as the current situation is concerned, it is still difficult for us to make up our minds to cancel some contents. For the sake of prudence, in this reform experiment, we kept the basic content and basic training in the traditional teaching materials.

We believe that it is not important to update the content of advanced mathematics at present. What is important is to avoid the tendency of tedious teaching and not to make a fuss about some small problems. Doing so not only distorts mathematics, but also makes students miserable.

Courand, an internationally renowned mathematician, once sharply criticized mathematics education. He pointed out: "Since 2000, mastering certain mathematical knowledge has been regarded as the intelligence that every educated person must possess. The special position of mathematics in education is facing a serious crisis today. Unfortunately, math educators should be responsible for this. Mathematics teaching has gradually become a meaningless exercise of simple calculus, which can develop the ability of formal calculus, but it is not helpful to truly understand mathematics and improve the ability of independent thinking ... "

Courant was right. Mathematics education needs reform, and we have a long way to go.

(The writer is a professor at Peking University Institute of Mathematics. )

Advanced Mathematics (2nd Edition), edited by Jong Li and Zhou Jianying, Peking University Publishing House, priced at 58.00 yuan (up and down).

Low starting point and strong sense of the times

Comment on "A Concise Course of Mathematical Modeling"

dirty word

Since 1990s, the word "mathematical model" has gradually become popular in middle school mathematics education. This is a great event to celebrate. In fact, the introduction of "mathematical model" into mathematics teaching not only adds a little teaching content, but also involves the change of mathematics concept, which promotes the progress of mathematics education as a whole.

For many years, China's mathematics education has tended to ignore the "mathematical model" teaching. Mathematics is equal to "logical reasoning" and "solving difficult problems", which highlights the rational thinking of mathematics, but denounces the application of mathematics and the establishment of mathematical models as "pragmatism" and "short-sighted behavior", and the result is bound to be one-sided. If this wrong view of mathematics is allowed to continue, mathematics will lose contact with reality.

Fortunately, this mathematical concept of ignoring application is changing rapidly. In mathematics education, mathematical modeling has become an integral part of national mathematics curriculum standards. Mathematical modeling contest and mathematical application contest have attracted a large number of teenagers to pay attention to the application of mathematics.

Professor Yuan Zhendong from the Department of Mathematics of East China Normal University, in order to promote the development of mathematical modeling teaching in middle schools, combined with years of teaching and scientific research experience, compiled a book "A Concise Course of Mathematical Modeling". This book has a low starting point and a small slope, but it covers the main contents of mathematical modeling, paying special attention to the elaboration of mathematical modeling ideas and the application of computers, and has a strong sense of the times. It is expected that through these efforts, students will learn how to abstract mathematical problems from reality, how to collect overall data, how to correctly use the learned mathematical knowledge and methods for re-creation and innovation, and use computers to obtain mathematical models that meet the requirements of problems.

A Concise Course in Mathematical Modeling, edited by Yuan Zhendong, etc. , East China Normal University Press, February 2002, pricing: 18.00 yuan.

The content is concise and classic.

Comment on A Concise Course of Mathematical Analysis

outer city wall

A Concise Course of Mathematical Analysis starts with the description of real number continuum, establishes a strict concept of limit, thus establishes the operation system and theory of calculus, and tells some basic applications. The transition from discrete operating system to continuous operating system provides a set of continuous operating system and its mathematical theory, so it is brand-new.

The contents of the book are all based on the real number system, and the four operations are closed and continuous, so it becomes the stage of mathematical analysis activities. In the process of learning this book, we should also pay attention to the combination of physical and mechanical background model (physical object), geometric image intuition (image) and abstract calculus reasoning (quantity), so as to form typical examples in our minds and calculate them skillfully and accurately. This book is also the most basic course for studying mathematics, natural science and modern science and technology.

This book is concise and classic, with some classic exercises, some difficult problems and some postgraduate entrance examination questions. This is a good book for students majoring in mathematics.

However, the book also has some mistakes caused by negligence. For example, case 4 ~ 5 on page 15 of Volume 2 should be un+ 1, but it is uu+ 1 in the book.

Generally speaking, this book is well written and worth learning by math lovers.

A Brief Course of Mathematical Analysis (up and down), by Deng Donghao and Yin Xiaoling, Higher Education Press, June 1999, priced at 27.40 yuan, 13096.