Teaching reform has two basic guiding principles: one is "demand-driven, application-oriented". According to the analysis of a large number of applications in the follow-up courses of electromechanical specialty, it is proposed that the goal of this course is to solve linear algebra problems of order 6 and above; Second, "technology promotes the introduction of computer computing", with the help of modern means, it realizes the combination of abstraction and image, the combination of written computing and computer computing, and the combination of basic courses and professional courses. We have specifically done the following work.
It comprehensively shows the teaching requirements of this course.
The requirements of the four modernizations for educational modernization are firstly manifested in the increasing expansion and deepening of the requirements for specialized courses, and then reflected from specialized courses to basic courses, thus promoting the reform and innovation of the whole teaching plan. In order to ensure a high level of teaching, this demand chain must be demonstrated frequently, and the ABET demonstration often carried out by foreign universities includes this content. Unfortunately, we haven't seen such a demonstration of linear algebra course in China, and it seems that no one cares about how the course content meets the needs of professional courses. When we do this work, we take a large number of electromechanical majors as the objects, analyze their needs in matrix modeling and calculation in the follow-up courses, and determine the tasks of the linear algebra course.
According to the compiling practice of scientific calculation in many courses for more than ten years, we find that there are more than ten courses in which linear equations can be used before the third year of these two professional universities: chemistry, advanced mathematics, circuits, theoretical mechanics, material mechanics, calculation methods, heat transfer, physics, computer graphics, signals and systems, digital signal processing, mechanical vibration, robotics and so on. (Examples of matrix modeling and calculation have been included in this version). However, in the actual textbooks, these courses basically do not use matrix calculation, because the content taught by linear algebra at present is out of touch with the needs of subsequent courses.
The first manifestation of disconnection is order n (referring to the large number of equations and variables). The order of chemical equation balance is the total number of substances before and after the reaction. There are three substances on each side, and n is 6. In statics, there are six equations for the balance of a single object in space. If there is one more object, the equation will be doubled. The number of nodes in the circuit diagram corresponds to the number of equations, so it can be seen that the n required for university courses is at least 6, but it will reach hundreds in engineering practice. The second manifestation is the type of equation. In physics experiments and various measurements, overdetermined equations are often encountered by using redundant data to improve accuracy, but most of the existing courses do not talk about overdetermined equations. The third performance is the number field. Complex algebraic equations are often encountered in AC circuits and signal processing.
According to the above analysis, we set the practical goal of this course in the teaching reform as: on the basis of maintaining the original theoretical and practical level, make students learn to solve complex overdetermined linear algebra problems efficiently. The difference between the old and new teaching requirements can be illustrated by charts. The white part of the figure is the coverage area of the current contour, and generally only the third-order problem can be solved, and only the eigenvalue can be obtained to the second order. The gray part is difficult to calculate by hand because of the complicated calculation, while the black area does not teach the algorithm at all. In addition, because it is limited to the real number problem, the white area is reduced by half. The new goal is that the problems solved by students can cover the whole world, including real numbers and complex equations, and the order can be extended to dozens or hundreds, so that it can be seamlessly connected with subsequent courses.
More than 80% of the topics in the follow-up courses will not fall into the white area, and there are probably few linear algebra teachers who can do this book, which can also show the educational level of linear algebra in China.
How to introduce computing software
Why don't people want to solve problems with matrices? Because there is no tool for matrix operation, even low-order problems are not as efficient as substitution method and elimination method, and people prefer to solve them by middle school method. As for the high-order complex number of n, it is even more sighing, so if you don't teach tools, you won't want to use matrices in subsequent courses. Many students report that linear algebra has never been used in undergraduate courses, and only postgraduate entrance examination is useful; This simply cannot reflect the position of linear algebra as a basic course in the teaching plan, let alone let students understand the important position of linear algebra in scientific calculation.
The only way to solve high-order complex equations is to use computers, especially the best software tools. In fact, not only computers are helpful to linear algebra, but also linear algebra is helpful to scientific calculation. The advantage of computer over calculator lies not in the single operation speed, but in the continuous operation and processing of massive data, and the best way to organize massive data is matrix. For example, to calculate the Fourier transform of a point 1024, it is necessary to multiply the signal array by a square matrix of 1024× 1024, which contains more than1000 million data (see the example in this book 10.9). Without the concept of matrix, how to assign values and how to place such a large amount of data is a problem. Therefore, it is difficult to give full play to the computer's ability before learning matrix, and it is the most suitable choice to complete the transformation from calculator to computer when learning linear algebra.
Some teachers object to the use of computers in linear algebra classes, on the grounds that when students use computers to calculate problems, they will inevitably relax their writing and not use their brains, which will affect their understanding of basic concepts. Some people even ban primary school students from using calculators as a reason to ban computers in the first year of college.
Teachers who hold the above views probably haven't worked out the application problems of order 6 or above, and they don't know what it's like to do multiplication by hand for hundreds of times without making mistakes, let alone the importance of CAD in modernization. Therefore, it is extremely harmful to education and science to be afraid of students being lazy and not teaching them advanced knowledge. I don't know how many times I have repeated the argument that "new technology makes people lazy" in history, but it just proves that I am "lazy" in the subsequent era. In the open information society, we should encourage college students to look for the latest technology to solve any problems from the Internet and around the world. How can we block knowledge? In a sense, if people don't want to be lazy, there will be no invention of various machines and scientific and technological progress. Being lazy by scientific methods is an act worth advocating. How can universities cultivate "innovative talents" if they do not strongly advocate learning the latest knowledge? Let students master more knowledge and calculation skills, and at the same time, in order to prevent them from swallowing dates, when doing questions and asking questions, they should turn around and let them use their brains instead of simply copying them. This is the art of teaching; But we must never just teach stupid methods and not teach new technologies. Otherwise, what are the "three aspects"? Isn't it an ignorant fool to compare the students trained in this way with the talents of the same level trained in developed countries?
In our teaching practice, * * * more than 800 students have participated in the teaching reform pilot. The students in the pilot are generally in high spirits because they have just entered the university and are proud that they can do both written and computer calculations. Not only do their practical problem-solving ability greatly exceed that of ordinary classes, but their theoretical test scores are also higher than those of ordinary classes. We don't pay special attention to theoretical education. The reasons for the improvement may be: (1) A large number of examples and visual teaching have improved students' learning enthusiasm; (2) advocate the combination of written calculation and computer calculation in the course; (3) The time saved by students in calculation is conducive to their more conceptual thinking. Practice has proved that it is unnecessary to worry about students being lazy.
We think the education department should really plan the whole process of cultivating students' scientific computing ability from primary school to university. The best way to prevent students from indulging in the internet is not to block them, but to guide them to learn and calculate problems with computers. "The computer should start with the doll!" Internationally, it is never too early for freshmen to learn to calculate problems with computers.
How to Cultivate Students' Abstract Thinking Ability
In the previous teaching of linear algebra, there were few application examples, and it was too complicated to use digital operations. Therefore, we have to take "abstract thinking ability" as the main training goal of the course, but we don't agree with this formulation. First of all, the goal is improper, the freshmen of engineering have little perceptual knowledge, the concept of three-dimensional space has yet to be established, the practical knowledge of engineering is almost blank, and abstract thinking has no foundation at all; Teachers put many unsolved practical problems in pictures or books, but they want to teach empty "abstract thinking", which is putting the cart before the horse and harmful. The second is that the method is wrong. Can we cultivate the ability of abstract thinking only by talking about theory without integrating with practice? Not at all! If students can't use matrix modeling to solve problems in their familiar curriculum fields, how can they be expected to think more profoundly in the abstract? This is the law of human thinking!
Teachers and students have to undergo a lot of training from sensibility to rationality in order to cultivate abstract thinking ability. Therefore, we should make use of the advantages of software tools to visualize abstract concepts in teaching; It is necessary to introduce a large number of matrix modeling examples to make students realize the advantages of using matrices. In fact, if you watch it too much, you will imitate it. Only by letting students see that linear algebra can be modeled in all courses they are familiar with and solve problems quickly can they gradually learn to use matrix modeling.
I have been exposed to matrices since the 1970s, but they are only used for reasoning and cannot be used for calculating problems. 1995 contacted MATLAB and found its special advantages in solving matrix problems. As long as the matrix expression is written, the problem can be solved quickly. So I tried to use matrix modeling and problem solving in each course, and wrote many teaching materials (see references) involving more than ten courses, so that no matter how complicated continuous and discrete signal flow diagrams are, they can be solved by computer conveniently. In this book, it is reflected in sections 8.6.3, 8.6.4 and 10. 1 1. This shows that abstract thinking needs to be based on a large number of modeling practices, and only by mastering advanced problem-solving tools can it be motivated; And it depends on long-term scientific research practice, not on a math class that only talks about theory without linking it with practice.
Combining science and technology with teacher training
Engineering linear algebra belongs to engineering mathematics. To apply mathematics to engineering, teachers should not only have a solid mathematical foundation, but also have the necessary engineering knowledge. When mathematical software is added to the teaching reform of linear algebra, teachers must master the programming of the software; In addition, there must be accumulation of teaching experience. Linear algebra is a wide-ranging course, with1~ 20 million students attending this course every year. There may be as many linear algebra teachers in China as1~ 20,000. Some are math majors, and some are engineering majors. It takes some efforts to meet the teaching requirements of both written calculation and computer calculation.
Some people suggest that only those who study mathematics can teach math well, which is very one-sided. In addition, in order to compete for the workload of their teachers, the engineering departments of some schools think that linear algebra is simple and easy to teach, and the new engineering teachers can cope with it without counseling and training, which is also wrong. Demand and application are the direction of discipline development. In order to better serve the application, engineering mathematics needs teachers who are interested in engineering as well as mathematics. They can all make great contributions to the teaching reform of engineering mathematics. On the other hand, a teacher who is from a mathematics major and is not interested in engineering, or a teacher who is from an engineering major and is not interested in mathematics, is definitely unable to teach engineering mathematics well. According to our experience, the combination of these two types of teachers is very important in the teaching reform of engineering linear algebra, and they should never have the same opinion. They should learn from each other's strengths. Our newly edited textbook "Engineering Linear Algebra (MATLAB Edition)" was completed with the cooperation of engineering and math teachers.
Foreign experience is worth learning. LACSG's five suggestions on the teaching reform of linear algebra in the United States were jointly put forward by mathematical experts and engineering authorities. Engineering mathematics teaching in American universities is also undertaken by engineering departments, such as probability theory and mathematical statistics courses in Stanford University. Even for engineering mathematics taught by professors of mathematics department, their engineering knowledge is generally much better than that of domestic teachers because of the late division of university majors and the combination of scientific research projects with engineering, which can be seen from the rich engineering background in their linear algebra textbooks (see reference [1]~[4]). In order to push the curriculum reform in the right direction, it is necessary to strengthen the training and evaluation of teachers. No matter what the background, teachers should have great interest in engineering and mathematics and be willing to combine them better. The training, assessment and selection of teachers should be based on four conditions: solid mathematical foundation, necessary engineering knowledge, good programming ability and rich teaching experience.
Problems needing further discussion
In order to test the effect of teaching reform and cope with the postgraduate entrance examination, our reform is bound, that is, to keep the original theoretical content of linear algebra unchanged, only to increase the practical content and improve the level of problem solving. This is bound to increase class hours. Among the additional credits, MATLAB accounts for 4 hours, computer time 10 hours (5 hours), and linear algebra practice accounts for 6 hours. If 9 hours of MATLAB is not counted, linear algebra takes 6 hours more, including the solution of overdetermined equations and additional theories related to calculation (such as calculation speed and accuracy, condition number, singular value decomposition, etc.). ).
In fact, there is indeed room for reduction in the hours of linear algebra theory. LACSG recommended that the whole course should be oriented to non-mathematics majors and highlight its application; No longer emphasize abstract thinking, only open another course for the department of mathematics to strengthen abstraction. Some universities simply rename engineering linear algebra as "matrix application", and these measures are all aimed at reducing the abstract "mathematical taste" of the original linear algebra class. In my opinion, it should not be difficult to reduce the theoretical part by six hours if the examination questions of the postgraduate unified examination can be reformed synchronously, reduce the abstraction and highlight the application. Of course, this still needs to be explored by everyone, and it also needs the correct guidance and intervention of teaching guidance departments such as the teaching Committee and administrative and examination departments.
The evaluation opinions of the teaching Committee and experts of our school on this project.
In May, 2008, an appraisal group was jointly organized by the Steering Committee of Mathematics Basic Course of Mathematics and Statistics Teaching Committee of the Ministry of Education and xidian university, and the project that has been implemented for many years was accepted and appraised. The appraisal opinion pointed out that:
"This project examines the teaching contents and methods of linear algebra from the perspective of engineering technology application, and integrates engineering background, application examples and modern scientific computing software into linear algebra teaching, which conforms to the direction of teaching reform at home and abroad and the international trend, and helps to achieve the goal of' improving the level of educational modernization', which is the first in domestic linear algebra teaching.
The two teaching materials, Practice of Linear Algebra and Introduction to MATLAB and Engineering Linear Algebra (MATLAB Edition), compiled by the research group, well reflect the combination of classical theory and modern calculation methods, visualize abstract concepts, realize some complex calculation problems, stimulate students' interest in learning, cultivate their ability to solve problems, improve the teaching quality, and lay a good foundation for the application of linear algebra knowledge in subsequent related courses.
The reform ideas and experience gained by our research group are exemplary, and the published teaching materials and teaching practice have had a great influence inside and outside the school. The effective teacher training courses organized by our research group and the compilation, courseware and other teaching materials provided have created good conditions for the popularization and application of this project.
The expert group spoke highly of the remarkable achievements made by the project team in the teaching reform of linear algebra over the past two years, and agreed that the project reform concept is advanced, distinctive and innovative, which is a high-level teaching reform achievement and has good promotion value. "
We will better implement the opinions of the expert group and strive to promote this achievement throughout the country. The revision of this book is one of the measures. I hope more universities and teachers in the country will participate. The United States has spent six years promoting the project of "improving the teaching of linear algebra with software tools" throughout the country, but it is difficult for China to do this in the same time without great efforts. Whether the software tools are learned in linear algebra class will directly affect the modernization process of almost all subsequent courses, and of course it will also affect the modernization process of education in China.
It is also an important aspect to provide on-the-job training for those who have studied linear algebra theory, including senior students, teachers and engineers, otherwise they will not be able to solve practical engineering problems. In order to better meet these readers, we added the chapter 10 in the second edition of this book, and added some in-depth examples of linear algebra application in the subsequent courses of mechanical and electrical majors, but junior college students don't have to learn this chapter.
Publication instructions
Because of the combination of computer and linear algebra, there are some new problems in the printing and typesetting of this book, which need better integration. We made the following treatment:
(1) In the narrative, according to the typesetting rules of the original linear algebra book, even if MATLAB functions or statements are encountered, the matrix is still in black italics, and the subscripts are still lowercase letters, such as [p, lambda] = EIG (a3).
(2) All human-computer interaction parts in the book are white. That is to say, all MATLAB complete program segments or program lines input into the computer are white, for example, A 1=A3*A4, because the computer does not accept black italic matrices, and small characters cannot be used for subscripts. This is consistent with the download assembly we provided. After the program runs, the results displayed by the computer are all white. For example:
Input [p, λ]= EIG(a3)
get
(3) Except for the matrix order Am×n, the multiplication operator does not need×, and adopts * uniformly, or it is completely omitted, such as A*B or AB.
Chen was born in July 2008 in Silicon Valley, USA.