First, the significance of high school students learning mathematics
We have been exposed to mathematics since primary school, but many people don't understand what we have been studying mathematics for so many years, just for exams again and again. It is more serious to ask such questions when we are already high school students, which shows that we are still learning mathematics blindly, without motivation and initiative, just because the teacher says we want to study. There is a lot of mathematics knowledge in our life, and learning mathematics is also to better solve problems in life.
Moreover, the ingenious skills and application of mathematics deserve our efforts and study hard. Learning mathematics helps us to be diligent in thinking, good at learning from, good at using and brave in innovation. Learning mathematics is of great significance. In a word, mathematics can make us better. We need to learn mathematics to improve ourselves in many ways. Our thinking and life are inseparable from mathematical logic. Deeply aware of the significance of learning mathematics, we should take action and study mathematics firmly from now on.
Second, to learn high school mathematics well, we should use the correct thinking method.
It is very important for high school students to learn mathematics knowledge and master the correct thinking method. Among them, induction and summary are common thinking methods in mathematics. Usually, we can combine a common problem to find the existence of * * * between them and draw a conclusion. This method can deepen our memory and understanding. Because getting another question from one question can develop our thinking, better understand and master the difficult formula, and at the same time make good use of its formula.
There is also analogy, which is similar to induction. It's just that analogy should be able to find different problems from multiple things and understand and remember them by comparing their differences. These two methods are of great help to master our basic knowledge and improve our learning ability.
Third, to learn high school mathematics well, we should preview and review well and master teaching methods.
Before we start the math course, we must have a general understanding of the teaching content, that is, we should form a good preview habit before class. Preview is very important for our class. In the process of preview, we can write down the knowledge points that we don't know very well. In this way, in the process of teaching, we will concentrate and listen carefully, so as to better master knowledge. Of course, we should also listen carefully to what we have previewed in advance, so as not to miss the missing parts in the preview. If we still have problems in class, we should consult our classmates or teachers in time, otherwise we can't learn math well.
At the end of the course, we should fully do the questions, consolidate our knowledge and expand our thinking through doing the questions. The key exercises are in the textbook, because the exercises in the textbook are all studied and demonstrated by experts, which can contain the learned knowledge points and are relatively reasonable. Compared with other counseling books, this is very meaningful. Because textbooks are the best counseling books, we can master the knowledge, draw inferences from others, and expand the thinking of solving problems, so that we can continuously improve the learning efficiency.
Fourth, cultivate a good interest in learning.
Confucius once said, "Knowing is not as good as being kind, and being kind is not as good as being happy." This sentence points out the importance of learning interest. Only by knowing it, loving it and having great interest in it can we enjoy it, generate the motivation of active learning and effectively improve learning efficiency. Therefore, in mathematics learning, we should turn this perceptual interest into a rational process of mathematics learning and cognition. So, how can we cultivate our good interest in learning high school mathematics? Can be considered from the following aspects.
(1) In class, we should actively cooperate with teachers' lectures, listen carefully and improve our mental attention, especially to solve the problems encountered in our preparation before class. In the process of listening to the class, we should keep pace with the teacher, pay more attention to the teacher's mathematical thoughts, ask more why, and turn the teacher's evaluation of ourselves into a driving force to spur ourselves to study hard, thus further stimulating our interest in learning.
(2) When thinking about problems, we should pay attention to induction and sorting out, and tap our inherent mathematics learning potential.
(3) Many concepts in mathematics are abstract and difficult to understand, which is the main reason for our lack of interest in learning. Therefore, in the process of learning, teachers should learn to return mathematical concepts to nature and real life, because all disciplines are produced from real life. Only by bringing the concept of mathematics back to life can we grasp the concept of mathematics more accurately and stimulate our interest in learning high school mathematics.
Fifth, cultivate the good habit of learning high school mathematics.
(A) diligent thinking
Thinking is a basic ability in senior high school students' mathematics learning. In the process of learning mathematics, we should think while reading, and do problems while listening to classes. In the process of thinking, we should have a deep understanding of mathematical knowledge, sum up mathematical laws, and solve mathematical problems more flexibly, so as to transform the knowledge taught by teachers into our own cognition.
(B) diligent in reflection
After doing math problems well, we should often reflect and review, which will help us master the key skills of solving problems and get the ideas and methods of high school mathematics.
(3) Diligent in decoration
We should always sort out the knowledge and structure of mathematics, form a block structure and assemble it as a whole, such as tabulation, which can make the structure of mathematical knowledge more clear and clear at a glance, and often analyze the categories of mathematical exercises and sum up all the knowledge in the same knowledge method.
(4) be diligent in doing problems.
If you want to learn high school mathematics well, you must try to consolidate and familiarize yourself with the knowledge by doing problems. Secondly, being diligent in doing problems can also cultivate students' independent thinking ability and flexible application ability of mathematical knowledge; In addition, while doing math problems, you can also integrate different math knowledge, carefully examine the problems, think carefully, and learn to solve math problems in different ways.
It is of great significance to learn mathematics well, so we should firmly study mathematics well. Learning mathematics well is the need of quality education, comprehensive education for all students, improvement of our comprehensive quality and unremitting struggle. As high school students, our task is to thoroughly understand what we have learned, strive for more chips for our future, and find a higher starting point for our future. What's more, mathematics in life is equally fascinating, and it is also a kind of significance for us to combine the mathematical knowledge in textbooks with the mathematical phenomena in life.
Skills of learning high school mathematics well
1, understanding of mathematics
Mathematics is actually not a very mysterious and supreme subject. This is not God's will. Mathematics has its own history and development. Of course, there are also mistakes and shortcomings, which is exactly what mathematicians should do now. Last year, I read a book called "The Loss of Uncertainty in Mathematics" (the first promotion series is actually a part of the history of mathematics), which made me realize that mathematics, like physics, is also an empirical discipline, but it is more rigorous than other disciplines (I personally think that mathematics and philosophy can solve problems that other natural disciplines can't). Mathematics is just a collection of predecessors' wisdom about "one aspect", and we are learning this wisdom, not dead arithmetic. You can send it to find some knowledge about popular mathematics, find something you are interested in, understand the development of mathematics, and get some inspiration and even interest.
Step 2: Interest
Learning any subject in high school needs interest support to learn well, not to mention mathematics is the main threshold.
But from many people around me, they all know that interest is very important, but they don't cultivate interest or cultivate interest subjectively.
But I don't have much experience in this field, so I have to wait for the teacher to supplement it.
3. Mathematical thinking is very important.
Our teacher said: there are several major mathematical ideas in high school: function and equation ideas, classification ideas, transfer and transformation ideas, limit ideas and so on. (If there are any omissions, please ask other teachers to add them. )
I think this idea is broad and should not be limited to these five ideas. Every subject of mathematics has its own ideas, and there are more than five. High school teaching should not be limited to these few, but should let students see more other ideas. I think I know a little, but I can't express it in words because of my poor level. I think this idea should also vary from person to person. Everyone has their own thinking characteristics, which need attention, so they should not be the same.
Although it is difficult to grasp the train of thought, there are still ways to get it: accumulation, but this accumulation is not the accumulation of doing problems, but the accumulation of thinking and summing up at ordinary times. If your method is different from others' when you are doing the problem, you have to think about how my method is different from its own. Whose method is good? Why don't you think so? Which method requires less computation? Which thinking is difficult? ...... For example, when you are studying or summarizing, you come across a familiar mathematical knowledge point, such as the original knowledge point, so you should consider why these knowledge points are similar. Do they have any apparent or substantial connection? Can you put it together? Can the method be universal? ..... after considering these, it will be useful. Many mathematical methods of crossing branches in mathematics (such as trigonometric substitution in radical calculation) are borrowed from this. Of course, there are still many places you should like, which depends on your own exploration.
Thinking and summarizing are very important in mathematics, and the quantity of thinking determines the quality of your mathematical thinking to some extent.
4. "Mathematical Feeling"
English has a sense of language. Sometimes you will feel that an answer is just like a correct answer. There is no reason, and many times it is exactly the same. This is the sense of language. Similarly, there is something similar in mathematics, which is called "mathematical feeling" for the time being. When we see a problem, we have an idea without thinking, which is probably the so-called "mathematical feeling". "Mathematical Feeling" is pure experience and can be accumulated. This kind of accumulation is the accumulation of doing problems, but I don't advocate this method, because it is easy to make mistakes and forget, and it is impossible to judge whether it is correct or not. But it may help you at a critical moment.
In fact, not only mathematics, but also the whole science. But this topic is too big for me to say. It depends on whether you understand.
5. Basic skills
The basic skills I'm talking about here are generalized basic skills:
1, basic calculation (accurate, fast, this is the most difficult, do not believe you see my mistake because of carelessness)
2, multi-layer discussion (this is more troublesome)
3, dictionary sorting (it depends on whether you can)
4. Algebraic deformation
5. Factorization
6. Solving equations
7. Elimination of parameters (including elimination)
8. Solving inequalities and proving inequalities
9. Find the general term of recursive sequence (including sequence summation)
10, trigonometric operation
1 1, calculation and proof of plane geometry
12, function evaluation domain
13, vector
14, solving simple indefinite equations and integer solutions
15, mathematical induction
16, complex number calculation
17, derivation
That's about it. Basic skills are an empirical problem, which needs to be accumulated through ordinary practice. It is necessary and endless to summarize some tips and tricks. But you can't spend too much time on this because: except for the first three, these basic skills are not the best (because no matter how good your basic skills are, you will always encounter problems), but these basic skills can't be too bad, because whether you can solve some strange problems depends on these basic skills.
6, have the spirit of innovation, believe in yourself (above the average level of mathematics)
Innovative spirit is the source of mathematics development, and it is necessary to have innovative spirit if you want to learn mathematics well. The spirit of innovation has many aspects. For example, when doing a problem, I feel that the solution to the problem is more troublesome, and there may be a good way. You can try it yourself and see if you can find it. This is an innovative spirit. But the spirit of innovation has a premise, that is, your math level can't be too bad. If you are innovative, you should dare to doubt. For example, the high differential method itself is not rigorous, so you can think about how to make it rigorous. Referring to mathematical analysis, I believe you will gain a lot (this requires you to have a certain mathematical foundation, which is broader than the basic skills of the previous point). In addition, the topic of mathematical analysis will be discussed later. )。 Innovative spirit can also play a role in solving problems. For example, a problem you have done has the value of promotion. You can try to promote it yourself, and then you can communicate with others (this requires you to have a broad mind, hehe, exaggeration), and then rethink your promotion and see where the problem lies. ...
But in the process of our innovation, we will always encounter difficulties. What should we do with them?
I think we should believe in ourselves from the beginning. Confidence is necessary. When I usually give lectures to others, I often encounter such a situation: a classmate tells me his story from beginning to end, and I nod all the way (a bit like typing all the way when installing a program), and he goes back with satisfaction without saying anything. This situation accounts for almost 50%. In fact, this shows that you don't believe in yourself. Mathematics is a very strict subject. Now that you have introduced it, there will be no problem, but if your foundation is poor and your reasoning is not rigorous, that is another matter.
In the process of exploration, you can't blindly believe in yourself, and it's easy to run into a dead end and waste time (hehe, there are risks to gain). This requires us to stop at an appropriate time, consult others, consult relevant books, enrich ourselves with the wisdom of our predecessors and save our time.
In the process of exploration, the most important thing is when to persist and when to ask for help. These two aspects have their own advantages and cannot be generalized. It is up to everyone to choose.
7. Expand knowledge (for students who have spare capacity in mathematics)
For students who are good at math, although the problems in high school are infinite, their thoughts are limited. These students should have mastered most of the ideas. However, there is still a lot of time before senior three. We can't let our existing mathematical thoughts and minds be abandoned (to some extent, it is a waste to do the problems we know), which requires us to expand our knowledge, acquire new ideas and understand new mathematical tools to keep the vitality of our mathematical minds.
My advice to students who have spare capacity in mathematics is to find what they want to see in the high school competition first (pay attention to what they want to see, because there are so many theories and methods in mathematics that it is difficult for a person to read it all his life). I think the following contents should be known to everyone:
Congruence, basic combination counting, pigeon hole principle, inclusion principle, basic parity analysis, etc. (to be supplemented later).
Of course, there are not many interesting places in the competition. Here are some branches of mathematics that you can refer to:
Mathematical analysis:
I think this is a must-read book for students who are good at math. Although you don't need to know, you must know that there is such a thing. You should know something about the meaning and application of derivatives and integrals. Integral is very useful, you will know after reading it.
Differential equation:
This branch is entirely prepared for physics. Anyone who likes physics can pass, but you need to look at mathematical analysis first.
Linear algebra:
There are determinant and matrix. I think everyone should know determinant, which mainly solves the problem of linear equations. Although the matrix is useless in high school, it is the only mathematical tool in mathematics that can accurately process a large number of data (at least I think so, and I think the future mathematics will definitely have greater development in processing a large number of data).
Combination count:
This thing is very interesting, but it involves a wide range and requires a wide range of mathematical knowledge. Among them, I prefer the method of solving symmetric problems (such as circular arrangement in space, or even more complicated problems) with permutation groups. You can also have a look if you are interested.
Graph theory:
I have to admit that graph theory is the most brainy branch of mathematics (in the branch I am looking at now). Although only addition, subtraction, multiplication, division and matrix are used, it is difficult to understand and is the best tool to practice thinking.