High school mathematics is characterized by abstract concepts, many exercises and high teaching density. Therefore, after the first year of high school, some students are afraid of mathematics.
In fact, it is not difficult to learn mathematics. The key is whether you are willing to try. When you dare to guess, it shows that you have mathematical thinking ability; And when you can verify the conjecture, it means that you have a talent for learning mathematics! If you really want to learn math well in senior two, it can also be understood as: how to make objects that meet the requirements with the least materials; How to allocate resources and put them into production to get the maximum profit; Why beautiful curves can be associated with algebraic equations; Why it is much easier to have a car accident than to win the sports lottery; Why do the same Amanome students often appear in each class in a year? ...
When you fall into the "trap" of mathematical charm, you have already taken the first step to learn mathematics well! 2. Learn to preview and listen to lectures
It's best to preview the contents of the textbook before class, otherwise there is a knowledge point that can't keep up with the teacher's footsteps in class, and the following is unknown. This vicious circle will start to get tired of mathematics, and interest is very important for learning. Targeted exercises after class must be done seriously and not lazy. You can also calculate the classroom examples several times when reviewing after class. After all, in class, the teacher is calculating and explaining problems, and the students are listening. This is a relatively mechanical and passive process of accepting knowledge. Maybe you think you understand it in class, but in fact, your understanding of problem-solving methods has not reached a deeper level, and it is very easy to ignore some difficulties that will inevitably be encountered in the real problem-solving process. "A good brain is better than a written one." For solving mathematical and physical problems, it is not enough to rely only on the general ideas in the brain. Only through careful written calculation can we find the difficulties, master the solutions and finally get the correct calculation results.
Review and summarize in time: In fact, whether you have completed the introduction or entered a higher level, another thing you have to do is to learn the basic knowledge well. That's all that matters. The basic knowledge of mathematics includes not only understanding definitions, memorizing formulas and using basic formulas, but also solving steps, considerable experience in solving problems and, of course, calculation accuracy.
Let's talk about it one by one:
(1) Understanding definition: Understanding definition is not memorization. I don't remember many definitions. Just understand. No one is asking you to remember the definition of something.
(2) recite the formula: needless to say.
(3) Application of basic formula: excluding flexible application.
(4) Problem solving steps: This should not be underestimated, and attention should be paid from the beginning. Steps are directly related to logic. With good logic, your steps won't be too bad. On the other hand, I haven't tried whether it is true or not.
(5) considerable experience in solving problems: this is the most important, but it is not a dead problem. Some problems, you can't, but you did, or did something similar, so that you can solve them according to the gourd painting gourd ladle, and you will be the same as you in terms of grades. Is it tempting?
(6) Calculation accuracy: Carelessness is also a non-intellectual error, which has always been a problem. In fact, I am also sloppy, sloppy for 5 years +4 years +3 years, and I have never solved it. I was inexplicably careless in the college entrance examination. But lucky people like me are few and far between, so don't take chances.
I believe that no matter how talented you are, you can do it. If you can't do it, it can only show that you don't study hard or have a wrong attitude or have other problems besides education.
We should be good at summarizing and classifying, looking for the * * * relationship between different types of questions and different knowledge points, and systematizing what we have learned. To give a concrete example: in the function part of senior one algebra, we have studied several different types of functions, such as exponential function, logarithmic function, power function, trigonometric function and so on. But comparing and summarizing, you will find that whatever kind of function we need to master is its expression, image shape, parity, increase and decrease and symmetry. Then you can make the above contents of these functions into a big table and compare them for easy understanding and memory. Pay attention to the combination of function expressions and figures when solving problems, and you will certainly get much better results. Finally, we should strengthen after-school exercises. Besides homework, find a good reference book and do as many exercises as possible (especially comprehensive and applied questions). Practice makes perfect, thus consolidating the effect of classroom learning and making your problem solving faster and faster.
Step 4 learn to solve problems
We know that learning mathematics needs to improve our mathematical ability step by step through review. Some students simply understand review as doing many problems, while others think that review is memorizing and reciting related concepts, theorems and formulas in textbooks. It can be seen that many students still have misunderstandings about review: they don't really realize the characteristics of mathematics, and they don't distinguish themselves from other disciplines in review methods.
Mathematics is a highly applied subject, and learning mathematics means learning to solve problems. It's wrong to engage in sea tactics, but it's also wrong to learn mathematics without solving problems. The key lies in the attitude towards the topic and the way to solve the problem.
-the first is to choose a topic, so that it is less but better. Only by solving high-quality and representative problems can we get twice the result with half the effort. However, the vast majority of students have not been able to distinguish and analyze the quality of the questions, so they need to choose exercises to review under the guidance of teachers to understand the form and difficulty of the college entrance examination questions.
-the second is to analyze the topic. Before you solve any math problem, you must analyze it first. Analysis is more important than more difficult topics. We know that solving mathematical problems is actually to build a bridge between known conditions and conclusions to be solved, that is, to reduce and eliminate these differences on the basis of analyzing the differences between known conditions and conclusions to be solved. Of course, in this process, it also reflects the proficiency and understanding of the basic knowledge of mathematics and the flexible application ability of mathematical methods. For example, many trigonometric problems can be solved by unifying angles, function names and structural forms, and the choice of trigonometric formulas is also the key to success.
-finally summarize the topic. Solving problems is not the goal. We test our learning effect by solving problems, and find out the shortcomings in learning so as to improve and improve. So the summary after solving the problem is very important, which is a great opportunity for us to learn. For a complete theme, the following aspects need to be summarized:
In terms of knowledge, what concepts, theorems, formulas and other basic knowledge are involved in the topic, and how to apply these knowledge in the process of solving problems.
② Method: How to start, what problem-solving methods and skills are used, and whether they can be mastered and used skillfully.
(3) Whether the problem-solving process can be summarized into several steps (for example, there are three obvious steps to prove the problem by mathematical induction).
(4) Can you sum up the types of topics, and then master the general solutions of such topics (we are opposed to teachers giving students ready-made topic types and letting students take topic sets, but we encourage students to sum up their own topic types).
Verbs (short for verb) enhance the ability of calculation.
The investigation of mathematics is mainly the basic knowledge, and the difficult problems are only synthesized on the basis of simple problems. So the content in the textbook is very important. If you can't master all the knowledge in the textbook, you won't have the capital to learn by analogy.
Secondly, we should be good at summarizing and classifying, looking for the * * * relationship between different types of questions and different knowledge points, and systematizing what we have learned. To give a concrete example: in the function part of senior one algebra, we have studied several different types of functions, such as exponential function, logarithmic function, power function, trigonometric function and so on. But comparing and summarizing, you will find that whatever kind of function we need to master is its expression, image shape, parity, increase and decrease and symmetry. Then you can make the above contents of these functions into a big table and compare them for easy understanding and memory. Pay attention to the combination of function expressions and figures when solving problems, and you will certainly get much better results.
Finally, we should strengthen after-school exercises. Besides homework, find a good reference book and do as many exercises as possible (especially comprehensive and applied questions). Practice makes perfect, thus consolidating the effect of classroom learning and making your problem solving faster and faster.
Do not use calculators at other times except when the square root, square root and trigonometric functions of general angles must use calculators. You must use a pen to calculate. Computing ability is the most basic and important requirement of science. The improvement of computing ability is not as immediate as learning English. It takes care, patience and long-term training. There is no shortcut. Insist on written calculation, not only in mathematics, but also in physical chemistry. When the topic involves gravity, electromagnetic problems or the algorithm of chemical equilibrium yield, it may be complicated, but manual calculation is completely feasible on the premise of retaining 3 digits. In addition, some simple and commonly used approximation algorithms should also be accumulated daily. It is worth mentioning that it is also feasible to calculate the square root with a pen, and its difficulty is similar to that of division with a pen. I suggest you consult a math teacher or a competition student.
6. Try some learning methods
Students with different learning levels need different learning methods.
If you are worried about the low state of mathematics learning, please follow the following requirements: after previewing, you can get twice the result with half the effort by coming into the classroom with questions; It is ignorance to want to finish beautiful homework, and it is more reasonable to correct mistakes; The exercise required by the teacher is not a "sea of questions". Please finish it carefully. If there is a genius who can learn math well with fewer words, it is not you. In the exam, the correct rate is as important as the speed of doing the questions, but giving up some questions reasonably can help you play your normal level.
If you are depressed because of the slow progress in mathematics, please accept the following suggestions: collect the wrong questions you have done, correct and write down the reasons for the mistakes, which are your personal wealth; For the test results, set yourself an acceptable bottom line and a goal within your power; Reasonable schedule and good study habits will help you get stable academic performance, so please make a good study plan and stick to it; If you spend a lot of time on a subject, you'd better allocate your study energy to each subject reasonably. People often have "plateau phenomenon" in the study of a certain knowledge field, that is to say, to a certain extent, no matter how hard they work, their progress will not be obvious. Mathematics focuses on cultivating the ability of observation, analysis and reasoning.
To succeed, learning methods play a vital role.
When learning mathematics, we must pay attention to strengthening learning flexibly, connecting with questions, analyzing and solving problems, using mathematical formulas flexibly, and not memorizing.
To learn mathematics well, we should first listen carefully in class, think deeply and explore the questions raised by teachers, and deepen and feedback the questions after class to ensure the consolidation of knowledge.
Moreover, the knowledge of mathematics is the widest, and there are many solutions to the questions, so it is impossible to read them in a short time. Therefore, learning mathematics requires "three hearts". That is, "confidence in learning mathematics well, determination and perseverance in studying hard." Only in this way can knowledge develop and thinking leap.
Because the problems in mathematics are ever-changing and complicated. We can't solve all the problems, so we don't need to do more when doing math problems. It is important to choose carefully and fully understand the type of a problem. By analogy, step by step, practice makes perfect. As the saying goes, "the sword front comes from sharpening, and the plum blossom fragrance comes from bitter cold", and the sweat paid is bound to be satisfactorily rewarded.
7. Cultivate scientific thinking
If you are depressed because of the slow progress in mathematics, please accept the following suggestions: collect the wrong questions you have done, correct and write down the reasons for the mistakes, which are your personal wealth; For the test results, set yourself an acceptable bottom line and a goal within your power; Reasonable schedule and good study habits will help you get stable academic performance, so please make a good study plan and stick to it; If you spend a lot of time on a subject, you'd better allocate your study energy to each subject reasonably. People often have "plateau phenomenon" in the study of a certain knowledge field, that is to say, to a certain extent, no matter how hard they work, their progress will not be obvious.
In fact, mathematics is not a subject of knowledge and experience, but a subject of thinking, and high school mathematics fully embodies this feature. Therefore, mathematics learning focuses on cultivating the ability of observation, analysis and reasoning and developing learners' creative ability and innovative thinking. So in the process of learning mathematics, we should consciously cultivate these abilities.
If you do the above two points well, then you can start to cultivate scientific thinking (or mathematical thinking). But in fact, no one did. After all, nothing is absolute, just like the first and second steps are mixed with the cultivation of mathematical thinking. Don't stick to theory, practice is the most important thing.
Further down is to improve scientific thinking, which is also very important. For more information about senior high school mathematics learning methods, please visit Tian Tian Learning Network.
Of course, this is all orthodox mathematics education. However, there are some differences between Guangdong education and orthodox education, that is, the requirements of Guangdong college entrance examination questions for science thinking (or mathematical thinking) are decreasing. I didn't want to accept this fact at first, but on second thought, the education here is mass education, not elite education. What's the use of doing a problem that only two people in the province can do? Where have you been shouting for so many years? It's right here. Before this year's college entrance examination, I guessed that this year's college entrance examination questions require less scientific thinking (or mathematical thinking), but many people don't believe that this year's college entrance examination questions have an answer. In this way, we seem to have a shortcut. If you can't reach scientific thinking (or mathematical thinking), then you can make up for your lack of thinking with the experience of doing problems. Of course, there is nothing you can do about it. If we can cultivate scientific thinking, we should still take the right path. After all, college needs it.
On the other hand, people who think too strongly in science can have a rest, after all, they don't take the college entrance examination. If your powerful science thinking finds some problems, you might as well play dumb. After all, the college entrance examination questions are not designed for people like you.
The relationship between learning methods and grades can be described as follows: when you are willing to understand the answers to most questions, your exam results should be easy to pass; When you are keen on studying all kinds of questions and making regular summaries, you must be an excellent student in class mathematics; And when you are used to doing your own problems and solving them according to the definition of mathematics, your mathematics level can already keep pace with the teacher!
Eight. Master mathematical thought
When your mathematical level reaches a certain level, you can enter the realm of learning mathematical thoughts. This realm is that different people have different views. Your level is probably higher than mine, and it may be useless for me to say it. But in any case, after completing the last stage, you can get full marks in the current college entrance examination questions, so keep going and do whatever you want, in short, you won't take the college entrance examination.
But there is one thing I can't help but say, symmetrical thinking. Symmetry is one of the common concepts in high school, and it is also one of the concepts that many senior teachers often ignore. Let's stop here. In fact, it's not that the teacher doesn't care, but it's not deep enough. Many problems have many strange symmetric solutions, and sometimes a difficult problem can be solved in a few seconds, so I put it here. Other teachers with better ideas will talk, so I won't talk about it. This symmetrical idea is so profound that I can't give an example because of the space problem (I don't have it at hand). If someone in your class sometimes uses words like "equivalence", "rotation" and "after the same …", because he probably uses a symmetrical method. If the teacher doesn't come up with that method, then this student is probably a master of symmetry, which is really a rare thing, even more difficult than winning the 5 million lottery.
9. Treat the exam correctly
College entrance examination requires advanced mathematics. If you want to get high marks, you must have a good attitude and write well (this is not a joke, of course, you should use a font similar to block letters instead of running script). These two points can be equal to mathematics itself in the examination room.
But in any case, your own efforts are the most important. Of course, it's not how late you study and how many problems you can solve. It's like going to a place without a road. You must go your own way, no one else can replace you. This is really a difficult problem, but if it weren't for this, I wouldn't be writing these useless theories here, would I?
The investigation of mathematics is mainly the basic knowledge, and the difficult problems are only synthesized on the basis of simple problems. So the content in the textbook is very important. If you can't master all the knowledge in the textbook, you won't have the capital to learn by analogy.
Ten: Three stages for senior high school students to learn mathematics well Many students and parents often ask me this question: "How can we learn boring and difficult mathematics well?" I feel funny, because in my eyes, mathematics has never been boring and difficult to understand. On the contrary, mathematics has an indescribable beauty, both beautiful and deep. This kind of mathematics has captured my heart and haunted me.
Mathematics is also a practical tool discipline. For example, the winners of the Nobel Prize in Economics are basically mathematicians. It will accompany us silently for a lifetime. If we don't get on well with it, it will be difficult for us to do anything. When talking about mathematical research, Mr. Hua mentioned three realms: 1, imitating with gourd painting gourd ladle; 2. Solve new problems with ready-made methods; 3. Put forward new ideas, create new methods and open up new research fields. This is also very enlightening for the study of mathematics. I think the realm of mathematics learning can also be divided into three stages:
The first stage: horizontally, the hillside becomes a mountain peak, with different distances.
Many students learn mathematics after understanding the teacher's questions in class, and then do the questions immediately. When they encounter something they can't do, they take out their books and open them, then do the questions again, and so on. As a result, if you encounter similar problems again, you will still be helpless and unable to start. Why? The study of mathematics is different from other disciplines. To truly understand the mystery, we should first pay attention to every definition, theorem and formula in the book, not just the conclusion. After careful study, it is the golden stone that inspires a method of solving problems. Therefore, when learning mathematics, it is suggested to thoroughly understand the connotation of the book first, which is the basic concept that must be tested in the college entrance examination, so as not to "know the true face of Lushan Mountain".
Stage 2: But you have broadened your horizons by going up a flight of stairs.
Understanding is not equal to learning in mathematics learning, which is a misunderstanding of many students. Understanding is only to understand the teacher's problem-solving ideas, and real learning is not only to correctly understand the teacher's problem-solving diagram, but also to sum up a method from the teacher's ideas for their own use. Some students learn mathematics only by completing the teacher's homework and are content to follow the teacher's footsteps. They pick up what the teacher has discarded and don't make any improvement. Gradually, they will close themselves in their own circles, making it difficult to activate their thinking. So it is safe to say that it is difficult to learn math well. Only students who walk in front of teachers and always leave enough room for improvement can leap to a new level with their own strength!
The third stage: looking back suddenly, but under the dim light.
Students and parents often say, "Why have I spent so much time and done so many problems in mathematics, but my grades have not improved?" What is the cause? I think this is also a problem that puzzles many people.
First of all, the problem is the problem. Some students and parents see that their math scores are not good, so they immediately go to the bookstore to buy a bunch of problem sets and start doing them. After doing this, they did that one after another, trying to win by the number of questions. This is not right. A good problem set has its own knowledge structure, and there will be a gradual process from shallow to deep, from a single knowledge point to multiple knowledge points, that is, gradient change. The problem is too complicated to be systematized, and it is difficult to form a gradient and cover. Therefore, when doing exercises, we should first carefully choose exercise books, and books with low quality would rather be discarded. Once a workbook is selected, it should be implemented. Be sure to start work. In the process of starting work, you can not only find hidden problems, but also concentrate on thinking. Many students learn mathematics without hands-on, which seems to have taken a long time, but in fact the effect is very poor; We must grasp the mistakes and not relax. The emergence of mistakes is the exposure of problems, and correcting them will raise a level. Therefore, when learning mathematics, you should be willing to spend time correcting the wrong questions. In a sense, it is enough to do this kind of exercise book well in one subject.
Secondly, the problem lies in thinking. The tactic of asking the sea doesn't work, but there are still students and parents who are keen on it. This is also wrong. There are too many math problems. When will it be finished? What is the concept of finishing math problems? Besides, there is no need to finish the math problem! In fact, math problems can be classified, and it is enough to do a few representative ones in each category. Therefore, people who can learn mathematics well are not only good at doing problems, but also good at thinking and understanding after doing them. One of the key points of this reflection is to classify the questions that have been done.
(This information is from Ai Xue. Jiangsu online, please indicate the source. )