First, rigor is the basis of rational thinking-"carelessness" is a bad excuse in mathematics learning.
As soon as the math exam results come out, students often sigh: "What's wrong with this topic?" And "I will do it, but I am careless." Hearing this, parents and teachers are often relieved and told not to be careless in the future, as if the problem had been solved.
In fact, no one wants to be careless in the exam. Everyone wants to finish the exam with high quality, but they can't avoid all kinds of mistakes. This is because the essence is not carelessness, but ability. The word carelessness masks many substantive problems.
I think carelessness is an improper classification of a large number of substantive problems. The so-called carelessness is generally not rigorous, and its subordinates are the defects of students' various abilities in learning. The operation is wrong because there is something wrong with the operation ability; There is deviation in understanding, which is a defect in understanding ability; Considering the problem is not comprehensive and the logic is not strict; Expression errors are problems such as expression ability. There are so-called carelessness in many links, but I don't think we can simply use the word "carelessness". We should realize that this is a problem involving all aspects of ability.
We should pay attention to the ability training in mathematics learning, the core of which is good study habits and rigorous consciousness. Let's take operation as an example. High school mathematics, every step of the operation, mostly the content of primary school mathematics, should not be wrong in theory. Many times mistakes are caused by students' inattention in operation, which leads to all kinds of low-level mistakes.
Of course, like carelessness, the problem of concentration is easier said than solved. People's concentration is often independent of their own will. Maintaining strong concentration in mathematics learning and problem solving is a kind of ability, which needs to be cultivated in daily training, and its foundation is good mathematics learning habits and rigorous attitude.
In mathematics learning, we should ask ourselves to concentrate on everything with a rigorous and serious attitude. This kind of high attention and all-out effort is a very important habit, which is the basis of improving ability and can form a virtuous circle of study, work and life.
Although I used to be an Olympic math coach, I don't agree that everyone should take part in Olympic math training. Students who have the ability and like to study, but have no foundation and don't like to study the Olympic Mathematics blindly will "muddle along". Obviously don't like it, don't want to do it, but force it, so students can't concentrate and be rigorous. Maybe students spend a long time, but the efficiency is not high and the quality is not good. Instead, it forms bad habits.
Second, questioning is the core of rational thinking-what, why and what else in mathematics education.
Mr. Zeng Rong, a middle school affiliated to Fudan University, summed up three things in mathematics learning, namely "what", "why" and "what else".
The content of mathematics in senior high school is more abstract than that in junior high school, and the concentration of knowledge is greatly improved. Many students suddenly find it difficult to learn high school mathematics because they have not adjusted their learning methods in time with the improvement of their ability requirements, and are used to learning high school mathematics with the learning mode of junior high school. This is a problem in learning.
So what's wrong with teaching? We often pay too much attention to learning or imparting specific knowledge and neglect to reveal the truth behind it. In some mathematics teaching, there is often no just a conclusion, the thinking process from condition to conclusion, and the link of questioning and reasoning is lacking, which compresses the thinking process of mathematics into a quick answer to the conclusion. Only pursue the speed of solving problems, not pay attention to the improvement of thinking quality. Be content to know what it is, and don't question why. Therefore, students' ability of questioning, induction and logical reasoning has not been fully cultivated, and the most effective opportunity to cultivate students' ability of questioning, questioning, induction and logical reasoning has been lost.
Mathematics learning should take students as the main body, and students can't learn passively. There are a lot of predecessors' creative work in high school mathematics, so we need to pay attention to the formation process of mathematical knowledge and concepts. The concepts of mathematical knowledge are all created by predecessors. Students can simulate the process of discovery and inquiry under the guidance of teachers, which is the most real innovation. For example, in solid geometry, why is the concept of angular size formed by straight lines on different planes defined like this? What kind of mathematical truth is involved? What kind of foundation has been laid for solving the problem? It seems that it takes more time to explore these problems carefully, but it can enable students to grasp the important information contained in the concept, tap the connotation of mathematical concepts, cultivate the spirit of questioning and exploring, and experience the simplicity and efficiency of mathematics.
Learning any subject should have the spirit of questioning. What we call questioning vision in mathematics is questioning what the essence of mathematical knowledge is, why it is like this, and what else. Only in this way can we finally promote mathematics learning and improve learning ability and thinking quality.
Third, interest is the cornerstone of cultivating rational thinking-simulating the history of mathematics and cultivating interest in mathematics learning.
Einstein said: "For everything, only love is the best teacher, far exceeding the sense of responsibility." I think, if you are not interested, you will never talk about "love". For a long time, we seem to have a general view that mathematics education in American primary and secondary schools is not as good as ours. But why, on the one hand, we think that the level of basic mathematics education in China is much higher than that in the United States, and on the other hand, many people question the role of mathematics education and hope that mathematics will "walk out of the college entrance examination"? The answer is actually very simple. If the purpose of mathematics education is to test and the process of mathematics learning is only to solve problems, then mathematics education is of course boring.
The interest of high school students is not simply based on fun, but more importantly, it makes students feel rewarding and enlightening. So what is the harvest? One view is to explain that mathematics comes from practice and can also be applied to practice, and it can be useful in life after learning. However, there are only a handful of high school mathematics knowledge that can be directly applied to life practice. Artificially fabricated "application" is not only unconvincing, but also one of the reasons why learning mathematics is futile. In fact, mathematics is the foundation of natural science, which is an acknowledged fact, and this is the most convincing application of mathematics.
In the process of mathematics learning, especially in senior high school, I object to one-sided emphasis on the relationship between mathematics and practical application, but I should expect to pay attention to the useless use of mathematics and think about mathematics education from the perspective of culture and human growth. The problem of mathematics education now lies in the organic connection between mathematics knowledge and the ideological culture behind it. There are only isolated knowledge points and topics, and there is no vivid process and experience. The emergence of any mathematical concept did not fall from the sky. The development of mathematics has both internal needs and external strength, which are very valuable mathematical resources.
Therefore, the charm of mathematics lies in letting students know the necessity and possibility of the emergence of mathematical concepts in textbooks, guiding them to relive or simulate the occurrence and development of these problems, allowing students to experience the joy of exploration and innovation while accumulating knowledge, and to be inspired by the background and corresponding methods of previous people's research on problems, thus realizing mathematical culture.
Fourthly, mathematics is a culture of rational thinking-why study mathematics?
"Mathematics is useless" and "the topic is difficult" are many people's impressions of high school mathematics, and some people even send out the voice of "Mathematics is out of the college entrance examination" on the Internet. So why does everyone have to learn mathematics that seems to have nothing to do with daily life from primary school, and many people forget it after the exam?
We might as well think about this problem from another angle. In China, probably few people ask what is the use of reading Tang poetry. This is because people all agree that Tang poetry is the traditional culture of China, which contains important educational value and cultural heritage, which is called "useless use".
We read Tang poetry since childhood in order to learn and feel the culture of the motherland. Similarly, learning mathematics is also learning a culture. Mathematics is a kind of world culture, and mathematics education also has educational value and cultural inheritance.