First, the process of cultivating thinking ability should not be rushed, but we should not miss such a golden age. It's like taking out the glass before it reaches the specified temperature, and you can't blow it if you want. But once it is taken out, it is very fast until it cools down.
Secondly, every child grows up like a molded glassware. How many things a glassware can hold depends on its capacity. I like to call this word measurement. How high a child can learn mathematics also depends on his thinking ability. Thinking ability is like the scale of a glassware.
The smart rabbit cited another problem that many parents will encounter in their lives. They always say that in junior high school, children are particularly good at math, and their scores in the senior high school entrance examination are above 145. However, after entering the high school stage, it went from bad to worse, and finally only got 890 points in the college entrance examination. The perfect score of the college entrance examination is also 150. Why?
I think the main reason is that children's thinking ability is basically stereotyped after entering high school, especially in senior two and senior three. If children's thinking ability only reaches this level, then the knowledge system in senior high school requires much higher thinking level than that in junior high school. Therefore, when you are so open-minded, you will want to memorize more knowledge systems and more content, which is absolutely beyond your power. In fact, this phenomenon has been rising slowly from the beginning, and it has already appeared at the beginning of Xiao Sheng.
Many parents gave me feedback, and my children never had to worry about primary school mathematics, and they all got 90 points 100 points. After junior high school, why did you get 70 or 80 points or even fail? In fact, the reason is the same. When a child's thinking ability can't keep up with the requirements of this knowledge system, the results will definitely not come out. Of course, parents don't have to worry. It is still the golden age to cultivate thinking ability. If you can seize this period and cultivate this ability, then the problem is not very big. But if you go to junior high school and feel that you can't learn anything, it will be more troublesome. Today I will talk about how to cultivate thinking ability from three aspects.
First, the cultivation of thinking habits, what are the good thinking habits;
Second, how to cultivate these thinking habits;
Third, from different stages, we should focus on cultivating different mathematical abilities.
First of all, there are many factors that affect children's mathematical thinking ability. For young children, the most important thing is to develop good thinking habits.
Let's talk about what is a habit of thinking. The definition in pedagogy is that thinking habit is a way of thinking formed through repeated practice, and it is the product of long-term accumulation and repeated reinforcement of conditioned reflex. It has relative stability. So what is the importance of thinking habits?
Knowledge and theory are like a set of kitchen utensils used by a chef, and the way of thinking is like a chef's cooking. Whether a chef can cook delicious dishes depends on his cooking skills or whether he uses this set of kitchen utensils. Then I think it must be the former, otherwise it is not a chef, then it becomes a blacksmith. A chef's cooking skills will also affect whether he can learn to cook a new dish as soon as possible, and even affect his ability to develop a new dish.
This ability, in my opinion, is a kind of thinking habit for children to learn mathematics. Therefore, the habit of thinking will be something that will accompany us all our lives. It is very important to develop good thinking habits and improve the quality of thinking. Thinking quality generally includes four parts: profundity, agility, flexibility and originality, which makes students' curiosity and thirst for knowledge about mathematics continue. In learning activities, gain successful experience, exercise the will to overcome difficulties and establish self-confidence; Experience exploration and creation, and feel the beauty and fun of mathematics. Savor these four sentences with the wisdom rabbit.
First of all, you should let your children start learning math. I think you should first have good thinking habits and improve the quality of thinking. This is the beginning of establishing habits and improving quality. After this beginning, the teacher will often tell you that interest comes first in learning anything. And here, interest is really the most important. Before cultivating interest, we must first cultivate good habits. With good habits, he can gradually develop curiosity and thirst for knowledge about this subject or thing in the process of repeated cycles, and gain some successful experiences from it, which is what we call interest.
With interest, with the deepening of his study, what he is bound to contact will become more and more difficult. At this time, he needs some will to overcome difficulties, including the confidence to persist in learning. This is the persistence stage. Once he has passed this persistent stage, I think he can really feel the beauty and fun of mathematics. Only after going through this whole stage did I think it was a very successful case of really learning mathematics well. You can find that all this, whether it is interest or persistence, and finally to experience the inner beauty of this matter, stems from his habits. Then let's talk about it in detail. What are the more important things about thinking habits?
What do good thinking habits include?
First of all, concentration. When thinking about math problems, you must be very focused and not have distractions. Now some children may turn on the TV when they go home to do their homework. Now they are more likely to watch their mobile phones or think while listening to songs. In this way, I think it is very unfavorable for mathematical thinking or thinking about mathematical problems.
Of course, I don't rule out some people. Music can make their thinking more exciting. Of course, this is limited to some light music or relatively soft melodies. It must not be some popular songs or songs with lyrics. Then as a bystander, as a parent, if the child is fully engaged, he must not interrupt his thinking.
When a person is completely immersed in thinking about mathematical problems, he will actually be very angry if he is interrupted. Because mathematical thinking is very rigorous and coherent. Once interrupted, it is likely to need to start all over again. If you disturb the child's thinking process, it will also have a negative impact on the persistence of children's concentration.
Sometimes I find that even if children go to physical education class and play basketball outside for 40 minutes during exams, they will occasionally make a loud noise. Even during the exam, some children, as long as they hear a little movement, their first reaction is to look up out of the window. You know, this is in the exam. What does this mean? Usually, when a child is doing homework at home, he must not concentrate on thinking, and there may always be something that will interfere with him.
Therefore, if you don't pay attention to this point at ordinary times, you can't concentrate on the exam. So that's why many parents give feedback: my children seem to be out of shape when they take the exam, or they don't do well every time. Concentration is a major reason for this situation. By the way, many parents are worried when their children grow up. My children seem to know everything they have learned, but they always seem to make mistakes in their calculations. Simple questions are always wrong, and we usually attribute them to carelessness. In fact, there are deeper reasons behind carelessness. Then I think a large part of it is when doing homework and exams, and the child's concentration is far from what he needs.
Second, a good habit of thinking is the habit of thinking independently.
For example, teachers in our school pay great attention to leaving enough time for children to think in class and discuss in groups. This thinking process is the most rewarding part for every child. After every child is involved, it will not only help to develop the habit of independent thinking, but also ensure the attention of the whole class. However, the practice in the classroom is still limited. Children actually have more time after class.
Therefore, our parents should encourage their children to think more independently, and don't always turn to some search tools or some developed means of communication, which are not conducive to cultivating children's habit of thinking independently and making them over-dependent. In fact, we feel that what we experience is really different from what others tell you. Even if your child is only aware of one tenth of this problem, as long as he has a little idea of his own, I think it is more gratifying than his imitation of others. Children's own gains will be even greater. This is the habit of thinking independently.
Third, the habit of observation. Observation is also a very fun thing. We often let children observe in class. Please observe the characteristics and laws of the conditions in this question. Then you will find some children looking at your blackboard intently, but you can tell at a glance that his eyes are dull. At this time, if you ask him what you have observed, he will only read the questions on your blackboard. This is what many children call observation.
This observation is tantamount to nothing. The real observation is to look at it purposefully, planned and systematically. First of all, children should be clear about what the problem is to solve, and then observe. What is systemic? A question consists of many statements and conditions. If this observation only looks at words, sentences and sentences, it is not called observation. You have to think, you have to think about the internal relationship between sentences. This is a systematic view, or a planned view.
There are many small ways to observe. For example, look for their differences on some very similar topics or things; Or look for the same thing or method in extremely different topics or things. We call it seeking common ground while reserving differences for short. Of course, this is seeking common ground while reserving differences in mathematics, and it is an observation method.
The fourth good habit is not limited to solving problems. We should explore the best strategy and study the same problem from different angles. In the process of guiding the competition, I often find that children are often only satisfied with finding the answer to the question, but ignore the thinking process of finding the answer. Is each step of your process rigorous enough, optimized enough, and can be reinterpreted by other methods and knowledge systems? In fact, these issues are very worthy of consideration. And a divergent thinking ability will gradually improve when thinking about these problems.
In the process of learning mathematics, you will find that all the results are certain, right is right, and wrong is wrong. There is no ambiguity or half a mistake. But in the process of exploring the results, it can be ever-changing. We often give a professional example. Children in grade six or seven can only learn to solve equations or inequalities from the angle of algebra.
The fifth important habit is to study problems from different angles.
Basically, teachers in Zhihui school will follow such a principle when giving lectures. Every lesson is not to teach children a certain knowledge point-this is the lowest thing in my opinion, but a process and a logical system to acquire this knowledge point. Only by putting this system into practice can we slowly cultivate children's ability to think independently and see problems from different angles.
The sixth good habit of thinking is the habit of writing and expressing.
It is mentioned in The Analects that learning without thinking is useless, and reading without thinking is useless. In mathematics, only solving problems without reflection will also affect the profundity of thinking. If you just solve the problem, then you just stay at the level of the problem itself. We often say that taking two steps and looking back at the road we have traveled will have a different understanding of life. Similarly, after solving a problem, you can naturally feel the gains by recalling the whole process of solving it, where are the key points, where are the highlights, where are not rigorous enough, and where can be improved. The deeper things behind the topic will gradually emerge.
This deeper thing may be a way or a feeling of being suddenly enlightened. Even we can experience a kind of values from a topic, which is a kind of mathematical beauty. Of course, reflection is not limited to the present. You can also advise your child to come up with some good problems that he has solved in the past, look back at past things, look at past ideas, and see if having a more mature knowledge system now makes you think differently.
Similarly, it is necessary to cultivate children's expressive ability. Besides writing, you should be able to speak. Whether you can explain your thoughts clearly also reflects whether the organization of children's thinking is clear. Let's talk about the importance of writing and expression from a professional point of view. Because mathematics learning involves the mutual conversion of several different languages, we generally call it three languages:
First, natural language, meaning expressed in words;
The second is symbolic language, which uses numbers and letters instead of written language;
Third, graphic language, especially how to write the proof process of geometry after learning geometry, or just an example I just gave, the image of a function, graphic language is also called image language.
How to switch these three languages naturally, and you can express yourself well in either language, which is a very important test for students' expressive ability. Expression ability is actually a habit of writing and expressing.
Seventh, the habit of checking.
Let's see, the habit of testing,
Let me take the game as an example again. Children who have participated in the competition say that they will encounter many competition questions to examine the optimal value of the function. When using matching method or basic inequality to find the maximum value of a function, some children will be happy to find out how much the range of this function is less than or equal to, and how much it is greater than or equal to, and write the equal sign as the maximum value of the function as the answer.
In fact, the trap of many competition questions is that this equal sign can't be taken. If your child has a strong sense of testing since childhood, then he will think of this more easily than other children. I'm going to have a test. After I find the range of this function, does this equal sign meet the conditions I get? As long as he thinks of this step, he can find this trap and solve this problem.
The above seven points are the seven most important thinking habits that I want to talk about to cultivate children's thinking ability. Review these seven habits again:
The first is to improve children's concentration;
Second, the habit of independent thinking;
Third, the habit of observation;
Fourth, the habit of studying problems from different angles;
Fifth, the habit of reflection;
Sixth, the habit of writing and expressing;
Seventh, the habit of checking.
Then why did the wisdom rabbit sum up these seven habits? I don't know if you found it when you listened to the lecture. In fact, as we mentioned just now, the quality of thinking includes four points: first, profundity, second, agility, third, flexibility and fourth, originality.
Let's look back. Among these habits just mentioned, the second habit of independent thinking is originality in thinking quality. Be sure to let the children put forward their own views and opinions, no matter whether this opinion is right or wrong. In fact, the result is not important. What matters is that he really thinks for himself.
The third observation habit, some students look at this question, they will know what laws there are between the conditions of the topic, or they can quickly see some clues. What is this embodiment? This is the embodiment of agility in thinking quality.
The fourth point is to study the problem from different angles, which is the embodiment of flexibility in thinking quality. Can I solve multiple problems? Then the fifth habit of reflection is the profundity of thinking I just mentioned. Looking back, I think I have gained something from this problem, which reflects the profundity of thinking.
How to cultivate good thinking habits?
Having said that, let's move on to the second topic, some concrete methods of how to cultivate thinking habits. Well, I'll make it simple, in two ways.
In the first aspect, it is not easy for students to develop good thinking habits. Because young children do not have strong consciousness, and their self-control ability and consciousness are relatively low, it is necessary to achieve the above good thinking habits through some necessary compulsory means or education.
There are usually three stages:
First, restraint, the appropriate coercive means I mentioned earlier;
Second, slowly and slowly let your child adapt to this constraint;
Third, let nature take its course.
I think we can start from the following aspects:
First of all, start with learning interest and motivation. Students' fascination with mathematics often begins with interest, and then generates motivation, and then from motivation to his willingness to think, and then from thinking to thinking success, a new round of interest and motivation is generated from the pleasure of success, which pushes this kind of learning forward continuously. Therefore, from the beginning, the cultivation of mathematics interest and motivation is the beginning of the cultivation of good habits. This requires us to give children appropriate support at the appropriate stage, or let them solve some appropriate problems and let them get it at once, so that a virtuous circle can be formed.
Why do you say that? Because many parents will give feedback, my child has entered a vicious circle. They have no interest from the beginning, and the less interest they have, the less they learn. The less he studies, the less he feels fulfilled and the less he likes it. I think it is very important to position children from the beginning. You don't want him to solve the problem in the first place. The difficulty you need to learn is to make him jump, make him feel that he has achieved himself, found some unique places of himself, or found some special skills of himself, and think that I can solve some problems. After he has this interest, he can gradually form a virtuous circle.
Of course, the second point here is to be strict. Strict requirements are divided into two points. Reasonable and moderate requirements conform to the law of children's cognitive development. At first, what is required of him must be realized through his efforts. Let him get spiritual satisfaction and encouragement after his efforts. Here I quote a passage from other teachers. I can't see the fox with the big red grapes in Aesop's fable. It wants to eat, but it is exhausted from dancing. Let's just say grapes are sour. The same is true of math learning.
Second, ask your child specifically. The following are the appropriate constraints. You must make a clear request. For example, as mentioned above, you made an agreement with your child that I would do math for the next hour and concentrate on math for this hour, which ruled out all other possibilities. In this hour, your mobile phone will be taken away, forcibly taken away. Of course, our parents try not to disturb the child for this hour, and then if he can concentrate for an hour, he will slowly extend it to improve the duration of his concentration.
Another example is the cultivation of children's independent thinking habits. There are too many assistive devices now. Universal search tools and portable communication tools absolutely hinder the cultivation of our children's independent thinking ability and perseverance. For children, those problems that can only be solved by jumping should be given enough time to think. Also during this period, he will never be allowed to use any external force or means.
If after this time, he has some ideas of his own, but they are not enough to solve the problem, let him seek other help. But today's children may think for a minute and then have no idea. They immediately searched for questions, discussed with their classmates and asked others what to do. That is the reality. Because the current means are too developed.
For example, to develop the habit of reflection, you can ask your child to prepare an exercise book, collect the questions he thinks are good, and tell him what this exercise book does, what things I should have, the correct problem-solving process and the reflection link. It doesn't matter whether he writes well or not, he must write something.
At this point, our school also has a sixth-grade teacher, who will ask the students in the class to write an article about their feelings after studying mathematics this week. I think this is a very good method and means, but also a reflection, that is, you can talk about the learning of knowledge points in the learning process. In your study, you didn't understand anything this week, but through unremitting efforts, you finally solved it. This is a harvest of perseverance and values that I just mentioned. This is a higher level of harvest, much stronger than harvesting knowledge.
Then, for example, the habit of writing and expression, writing is not standardized, I must ask for rectification and rewriting, and often ask children how to exercise their expression ability, why do you do this, and what conditions lead you to have this idea. So basically, I think we have put forward specific requirements for the cultivation of various habits.
What different mathematical abilities should children develop at different stages?
Thirdly, let's talk about the different mathematical abilities that children should cultivate at different stages. A child's thinking ability must develop with his cognition. In fact, the cultivation of various mathematical abilities should be gradual. From the perspective of competition, in the primary school stage, it is mainly to cultivate children's perception of numbers and the flexibility of mathematical thinking. This direction is very correct. Because the younger the children, the more they should open their hearts and come up with all kinds of interesting ideas and even some strange methods. Don't instill some methods and routines too early, which restricts the divergence of children's thinking. But this is my opinion.
Reality is on the other hand. For example, to be utilitarian, in order to win the prize in the competition, I have to solve the problem quickly. When many children first came into contact with Olympic Mathematics, they were more exposed to memorizing and imitating some technical problem-solving methods. Children never tire of it. Why? Because the skill-based problem-solving methods mentioned by the teacher are very good, I immediately solved the problem with a set of formulas. But why can you set it, why a setting must be the correct answer. I asked many children, and he didn't know why.
If we study like this and ignore the whole process of thinking, understanding and analysis, children will become problem-solving machines over time, which will hinder the creativity and divergence of thinking. Children who study in this way may win the first prize, the second prize in primary school and even the third prize in junior high school. In high school, you may not be able to participate in the competition. Because learning in this way, the higher you go, the easier it is to encounter thinking bottlenecks. Because your child didn't start from the perspective of developing thinking ability. Therefore, children's thinking must be liberated in primary school.
We often talk about liberating children's nature. In fact, learning math is the same. Let him do whatever he wants in primary school. It doesn't matter if he dares to think and say wrong. The process of daring to think and speak is the only process for children to improve their ability. After slowly entering junior high school, I will emphasize the cultivation of mathematical thinking ability, which will be systematic. With the perfection of knowledge system, it needs to be systematized. From the early stage to the first stage, the focus is on cultivating children's computing ability and abstract thinking ability from numbers to letters. In fact, this is completely consistent with the compilation of our teaching materials.
Question and answer; A
Ok, let's briefly answer some questions raised by some parents.
Q: Children's application ability is weak, how to solve it?
A: We find the following phenomena from the actual teaching: First, children don't understand many practical background problems in life. For example, in profit and loss problem, which is often tested in application questions, some children don't even know what the discount in shopping malls means, how to calculate it and how much the 50% discount is. When I encounter this kind of problem, I think it is actually related to what I just said and the cognitive level of children. Our education should match the children's cognitive level.
First of all, you should solve this problem and let him know all the practical problems involved in the application problem. In life, how do we calculate the discount on clothes? First of all, he can understand this problem and enter the stage of independent thinking. Second, children with poor application problems often lack the ability to think independently. Many children will depend on whether there is a teacher or not. Are there any tools to tell me how to list this equation, how to list it first, and then how to list it? This is not conducive to the breakthrough of application problems. Be sure to let him read the conditions in the application questions clearly first, and let him list each condition, and then ask him what each condition can bring you, whether it means a certain amount or an equal relationship.
Every condition in the application problem should have its function. In this way, separate each condition and let the children see one condition at a time. Of course, this is also a long process. From splitting the application problem to analyzing each condition, then combining the conditions of this application problem, and then asking him, how should our equation be listed? Maybe there will be some changes by exercising in this way.
Q: Do you have to learn Olympiad and take part in the competition to learn mathematics? Is there any other way to cultivate mathematical thinking?
A: First of all, you don't have to learn Olympiad to learn mathematics, and Olympiad is not necessarily suitable for all children. For a simple example, would you let every child learn 1 10 meter hurdles? Definitely not. In fact, the same is true. In fact, the cultivation of a whole set of mathematical thinking habits is not only aimed at children who learn Olympic mathematics, but also needs to cultivate these mathematical thinking habits and abilities.
To put it bluntly, mathematics is just a subject and a carrier now. Maybe he won't do some jobs related to mathematics when he grows up, but his thinking mode will accompany him all his life. Therefore, the seven points I just mentioned, whether you learn Olympic Mathematics or not, must be cultivated, that is, they will be cultivated in primary and junior high schools now.
Q: Children often jump around on math test papers. Should this be stopped? How to deal with it?
A: This must stop. As for how to stop it, some coercive measures may be needed as appropriate. Or you can communicate well with your children, especially those in adolescence and development, who are easy to rebel. You can communicate with him and listen to him. Why did he choose the topic and why did he jump? The general psychology of a child is that he basically chooses the questions he can do.
You can analyze with him, what is learning for? Learning is not practicing what you already know. I will still do it after practicing for a long time, but I will skip it if I can't. If you don't practice, you still won't. So you've studied for a long time and still know nothing. Communicate with your child first, then see how to deal with it, and analyze the specific situation.
The article comes from Sohu.