Introduction
As the day of college entrance approaches, the author has received a lot of letters from “prospective college students” who are full of expectations for the upcoming college life, but there is also a trace of worry: the university courses are difficult. Not difficult? Can university life be adapted? Especially as a mathematics teacher, what I hear most is the worries about the future college math courses. Because mathematics has always been one of the most difficult subjects, high school mathematics has left a lot of people's palpitations, and when I arrived at the university, I heard that I still have to study "advanced mathematics", "I am a person who has been fainted in elementary mathematics. Huh?"
This kind of worry is like a student who is about to enter high school after the entrance examination, and in fact, the gap between college mathematics and high school mathematics is far greater than that between high school mathematics and junior high school mathematics. So today, let’s talk about how prospective college students face the future college math courses.
What mathematics courses should be taken in university?
One of the biggest differences between universities and high schools is that they are taught by majors. Different majors learn different courses, and the same is true for math courses. Overall,
is roughly divided into 5 levels and .
- Grade 1: Mathematics Department of Mathematics
Mathematics Department regards mathematics itself as its own major, so it is also the purest and most difficult mathematics. Generally speaking, basic courses mainly include mathematical analysis, advanced algebra, analytic geometry , and more professional courses include real variable functions, complex variable functions, abstract algebra, differential equations, functional analysis, , and . The school will also learn more difficult professional courses such as differential geometry and point set topology . The names of these courses make people dizzy, and they kill countless brain cells. However, the author has written many articles about the study of professional mathematics, so I won't repeat them here.
- second grade: non-mathematics science majors
like physics, chemistry, etc. These majors are not mathematics, but also belong to science majors, and the requirements for mathematics are also very high. Some have reached almost the same level as the mathematics department, and at the same time they will also offer some mathematics courses that are emphasized in physical chemistry, such as group theory and .
- third grade: Engineering majors
Engineering majors are the main body of Chinese university majors, such as computer, electronics, communications, civil engineering, machinery , etc. are all engineering majors. The mathematics courses for engineering majors are what we usually call the three doors: advanced mathematics, linear algebra, probability theory and mathematical statistics . These three courses are also required for the corresponding postgraduate entrance examination. Some majors also offer courses such as complex function and integral transformation according to their needs. These mathematics must be mastered by engineering students, but the difficulty is less than mathematics major , do not pay attention to principles and proofs, but prefer calculations.
- The fourth file: Economics and management mathematics
This mathematics is for students of economics, management, business and other majors. The content is basically three doors: advanced mathematics, linear algebra, probability theory and mathematical statistics , but The content is further reduced and the difficulty is further reduced. A lot of theoretical knowledge has been deleted.
- Fifth file: Liberal arts mathematics
This is the lowest level of difficulty, for some so-called pure liberal arts, such as political science, literature, foreign language, law and other majors, basically belong to the nature of science. Only a brief introduction to some basic concepts in advanced mathematics, linear algebra, probability theory and mathematical statistics.
What are the main contents of mathematics courses in university?
We will only introduce the three major courses with the largest number of learners.
- Advanced Mathematics
Master of Advanced MathematicsThe body is calculus , but also a small amount of other content, such as space analytical geometry, differential equation and so on. Advanced mathematics is the most closely related subject in high school mathematics, because everyone has learned the part of derivative in high school, and derivative is actually the first half of calculus. Of course, the university will teach derivatives more in-depth and comprehensive than high school, not only to learn his principles and proofs, but also to learn a variety of new applications.
And the inverse operation of derivative is integration, and the combination of differentiation and integration is calculus. And in the research process, we need to use the new concept and tool of "limit" , so we must systematically study the limit theory, including the infinite series theory.
Many phenomena in nature can be explained by calculus
- linear algebra
linear algebra should be a rather unfamiliar subject for Chinese students who have only taken the college entrance examination. Its main research objects are vectors, matrices, determinants, linear equations and linear spaces , as well as their properties and their interrelationships. Many of these concepts were completely untouched in high school, so it is a relatively new subject for students. But this is actually one of its advantages, because it is a new research object, so there is not much high school basic knowledge required, so for those students who have a bad foundation in high school mathematics, it may be a good thing.
linear algebra mainly studies various spaces
- probability theory and mathematical statistics
probability theory is actually the content of high school, and students will not feel unfamiliar. But what you learn in high school can only be called classical probability theory , and when you get to university, you have to learn from the height of theory, which is the so-called axiom probability theory . From the perspective of set theory , probability theory is based on a solid axiom basis, and a series of concepts such as random variables, mathematical expectations, variance, correlation coefficient , etc. are derived from this. Of course, what the non-mathematics majors are learning is only the skin of axiom probability theory, and the real axiom probability theory can only be learned in the probability theory courses of mathematics majors.
is based on probability theory, but also learns mathematical statistics and . The two are related to each other but they are different. It mainly includes three parts: statistical distribution, hypothesis testing and parameter estimation.
What are the differences between college mathematics and high school mathematics?
The difference between non-professional mathematics and professional mathematics is still very large. Non-professional mathematics is basically a continuation of high school mathematics. It is nothing more than introducing new research objects, introducing some new concepts and calculation methods, and it is in the same line as high school mathematics in the core of thinking. The topics are still mainly calculations, and they are not very familiar with the essence and connotation of concepts. Therefore, they are still "question machine" -style learning. Of course, there is a reason for this. For non-mathematics students, the main purpose of mathematics is to use mathematics as a tool. Students only need to learn how to use the tool, or even use it proficiently, and use it to perform more complex operations. It is enough to solve practical problems.
But professional mathematics is completely different. Compared with high school mathematics, it is almost another subject. Or to be more precise, professional mathematics in college is the real mathematics. High school mathematics cannot be called "mathematics" (mathematics) , but can only be called "arithmetic" (arithmetic) .
So what is the difference between professional mathematics in universities? Mainly in the following aspects.
- 1. The two ways of thinking are different.
The mathematics in middle school and before is called elementary mathematics , while the mathematics at the beginning of university is called advanced mathematics . Of course, the advanced mathematics here does not refer to "Advanced Mathematics" This course refers to higher-level mathematics. So where is he tall? The main thing is the introduction of "limit" (limit), and "infinity" (infinity) These two concepts, and regard the limit as a dynamic process, not a result.
looks at mathematics concept from the viewpoint of dynamic change, which is not in elementary mathematics. Of course, in elementary mathematics, concepts such as "function" (function), , etc. will also be mentioned, and "independent variable" and "dependent variable" are also referred to, and the domain of the function can also Is the entire real number axis, that is, from negative infinity to positive infinity. However, elementary mathematics only stops at the study of various calculations and properties of functions, and does not discuss the nature of "change", "limit", and "infinity". One of the most obvious signs of higher mathematics is the use of dynamic viewpoints to thoroughly explore the essence of the above concepts and give a strict mathematical definition.
For example, the definition of "ε-N" about the limit of a sequence of numbers is to use any finite state to approach the infinite state, or to use dynamic finiteness to approach the static infinity. This is a subversive and revolutionary idea. It also brings a way of thinking completely different from elementary mathematics. The geometric interpretation of
limit
based on this, later developed a complete set of calculus theory, so it can be said that the complete set of calculus theory is based on this dynamic change of thinking. Each of the following concepts, such as continuous, derivative, integral , etc., are developed in this way of thinking. Therefore, if you want to learn college professional mathematics well, you must adapt to this new way of thinking as soon as possible.
- 2. The two research paths are different
Here I want to make a metaphor: compare mathematics to a big tree, high school mathematics and university mathematics are developed in two completely opposite directions. High school mathematics is like the branches of a big tree. Its task is to continuously grow upwards and grow more leaves; while college mathematics is more similar to the roots of a tree. Its task is to continuously grow downwards and go deeper. To advance in the place to provide a more solid foundation for the entire tree.
Specifically, many new concepts have been introduced in high school mathematics, such as sets, sequence of numbers, inequalities, vector , etc. We did not actually give him precise mathematical definitions, nor did we delve into their essence, but After roughly understanding their meaning, I started to do the questions, using various formulas to carry out complex calculations, and using various techniques to make subtle derivations, just like new branches and leaves growing on a tree trunk. Dao’s problems, calculations are becoming more and more complex, and skills are becoming more and more sophisticated, and the entire tree looks lush and lush.
However, the focus of college mathematics is not these. It will not pursue those whimsical skills, but will focus on the strict definition and essential meaning of these concepts, and constantly dig deeper into the connotations behind these concepts. Even the most basic mathematical concepts must be defined in the most rigorous language. For example, in subjects such as basic analysis and abstract algebra, we will study what is 1, what is 0, what is addition, and what is multiplication, and even let you prove that 1+1=2 and xx=0 , These questions are simply incredible for people who have not been exposed to professional mathematics, but in fact these are the most important questions in mathematics. Mathematics is the most rigorous logical system, and these are the foundation and beginning of the entire logical system. Only when the foundation is firmly established can the subject continue to develop. This is not available in all other subjects, and it is also a fact that all mathematics students should recognize clearly.
- 3. The problem-solving ideas of the two are different.
Problem solving is one of the most important steps in mathematics learning. Students familiarize themselves with formulas and understand concepts by doing exercises. However, there are obvious differences in the way of solving problems between high school mathematics and college mathematics.
The concepts in high school mathematics are relatively simple, but the topics are very difficult. In addition to various complicated calculations, various techniques must be used, such as scaling method , undetermined coefficient method, substitution method, number shape Combination method, split term method, design without seeking method, difference method and so on, all kinds of techniques are dazzling. And so-called problem-solvers or people with high IQ,It is to be able to quickly select the appropriate and correct skills from so many complicated skills, and finally get the result that the person who asked the question wants. I have to admit that this is indeed a high IQ performance.
and college mathematics is just the opposite of high school mathematics. Its topic skills are not so strong, but the concept is very difficult to understand. If middle school math problem solving depends on IQ, then college math problem solving depends on "insight" , see how well your understanding of mathematical concepts is in place and how profound. The topic of college mathematics is usually an investigation of the essence of the concept, and sometimes the topic itself can be regarded as a deeper understanding of a concept. Students do not have to worry about the complicated skills of high school mathematics, but should calm down, close their eyes and meditate. The deeper the understanding of the concept, the easier it will be to start the problem.
Although people with high IQ usually have keen insight, these two abilities are different after all, and sometimes contemplative thinking is better than flashing thoughts. Therefore, people who do well in high school mathematics may not be able to learn well in college mathematics. On the contrary, people who do not do well in high school mathematics may find themselves in college.
How to study mathematics in college?
is because of the above-mentioned differences, so the learning method of mathematics in college is completely different from that in high school. In high school, the textbook only provides an outline, and the real essence is in the tutorial book; while in college, it is the opposite, all the essence is in the textbook. Therefore, the most important method of college mathematics is to read textbooks repeatedly. Under normal circumstances, in addition to what the teacher teaches in class, you should read carefully and intensively at least twice after class. For some important or difficult concepts, you should read many times without passing a word. Every word And symbols must be carefully considered.
Of course, it is also necessary to do the questions, but different from the high school, it is not the pursuit of problem-solving tactics, nor the training of problem-solving methods and skills, but the deepening of the understanding of concepts by doing the problems. The core of college mathematics is the concept, and any exercises carried out without the concept are meaningless.
