How to teach mathematics? How to improve the efficiency of mathematics teaching, especially mathematics classroom teaching? How to help students learn mathematics better in teaching?
The "teaching" and "learning" of mathematics is a big problem, not the main idea of this book. Here only remind teachers and students to pay attention to a few problems that are easy to ignore. Some problems have been discussed in different forms.
(1) Learning Mathematics Reading
In primary and secondary schools, we will encounter such a situation. When students ask questions to teachers, some teachers often say: Please read the questions twice; Please talk about the problem; Please copy the question again; Wait a minute. What these teachers want to express is a meaning. Please read it again and understand it again.
We tell a true story. In universities, a "Mathematical Modeling Competition" is held every year. The problems in the competition are all practical problems. It takes three people to work for three days to complete a "paper" to solve the problem. You can use all kinds of books, network resources and tools (including computers and software). 1993 or 1994 Capital Normal University formed a team for the first time and asked us to be the instructor. We are very embarrassed. Students from Capital Normal University have to take the exam together with students from Peking University and Tsinghua. The gap is obvious and diverse. Our analysis shows that the biggest gap is the habit and ability to learn and understand mathematics independently. We changed the way of tutoring, let students choose the content, students say, we listen. At the beginning, we always said: Sorry, we didn't understand, please prepare again. Some students have said it four or five times, and when we think he really understands it, we will learn something else. This method is very good. After most students go through this process once, it will be easier to report other content. These students got good grades in the competition.
When learning a foreign language, you have a basic ability: reading comprehension. We think "math reading" is also very basic in math learning. In recent years, we have come into contact with the teaching practice of some primary and secondary schools, and there are fewer and fewer requirements and opportunities for primary and secondary school students to read mathematics independently. Teachers have good intentions. In order to improve students' test scores as soon as possible, to "talk more" and to "save time", teachers have done too much instead of students. We hope students realize that improving math reading ability is one of the basic skills to learn math well. We have done a survey, and in the papers of geological subjects, the number of mathematical formulas appears as many as 6 times per page on average. There are similar situations in other disciplines. In order to better illustrate the importance of mathematics reading in primary and secondary schools, we take mathematics "application problems" as an example to illustrate.
In mathematics teaching in primary and secondary schools, "application problem" is often a difficult point. Why is it difficult? There are two main reasons. One reason is that the background is rich, and they are all quadratic equations with one variable. But it can be displayed in various backgrounds, and it is difficult to define it as a question. If it is classified as "application problem of quadratic equation in one variable", it seems that it is not classified. It will be very complicated if it is classified from the background.
The second reason is that questions and conditions are not as standardized as traditional mathematical exercises, and sometimes it is necessary to define "required conclusions and conclusions to be proved" from narration. The relationship between "conditions" and "conclusions" is not as "decisive" as traditional mathematical exercises, that is, conditions are indispensable. In this way, it is necessary to analyze and judge which conditions are useful and which conditions are not, and the basis for analysis and judgment varies from topic to topic. These are not in line with the tone of primary and secondary school teaching-question type.
The application problem is "difficult" because it requires the ability of "mathematical reading comprehension", "difficult" because this ability can not be cultivated suddenly and can not be easily modeled, and "difficult" because teachers can not replace it.
Application problems, including mathematical modeling, have two educational functions. On the one hand, we can understand the relationship between mathematics and daily life, the relationship between mathematics and other disciplines, the role of mathematics in social development and the value of mathematics. On the other hand, from another angle, we can understand the process of doing mathematics. Mathematics is not only from concept to concept, from theorem to theorem, from some results to a new result; Mathematics has a background, which contains profound mathematical connotation and plays an important role in mathematical thinking. There will be a process in doing mathematics, which is a very interesting process. We need to find problems, guess, analyze and seek conditions, and we will constantly correct them, even repeatedly, and so on.
"Mathematics reading comprehension" ability is a basic ability, and both teachers and students should attach importance to it. It takes a long time to improve this ability, and teachers should provide different suggestions for different students.
There is a cognitive obstacle in mathematics teaching in primary and secondary schools. Some people think that "learning mathematics is doing mathematical exercises", while others think that "the ability to do problems is real, and everything else is empty." This view is reasonable, especially when dealing with exams. The ability to do mathematical exercises is an important aspect of reflecting mathematical ability. Doing exercises helps to understand some math skills and methods. However, the study of mathematics contains more contents, which we have talked about a lot before.
It is suggested that teachers should give students more opportunities to improve their "mathematical reading comprehension ability" for students with different levels and characteristics. Many teachers have accumulated some good experience in this field, for example, let students read textbooks and collect reference materials. In reading, let students think about the formation process of "some important concepts", the knowledge structure of some chapters, the internal relations of different concepts (such as functions and series), etc., and encourage students to write their own ideas into reports.
I hope students can broaden their thinking. Besides doing problems, they can also ask some questions worth thinking about and form the habit of thinking. When we were studying in the Department of Mathematics of Peking University, we asked Mr. Ding a question, which roughly means: What kind of students are good students? Teacher Ding's answer will never be forgotten. "Students who have no problems may not be good students." For many students, there is probably nothing worth thinking about except the exercises that they can't do. In the reading of mathematics, we should constantly ask questions to deepen our understanding of mathematics.
(2) Develop good math study habits.
In this curriculum reform, a three-dimensional goal is put forward, in which "process" is also regarded as a goal. "Study habit" embodies this process well.
What is a study habit?
Some students do their homework (usually exercises) when they come home from school, and even if they have finished, they have completed their learning tasks.
Some students, after returning home, first carefully read the textbooks taught by the teacher, then do their homework and think about what they learned today and what they learned before.
Some students have the habit of summing up. When learning a paragraph, they must sort it out and write it down.
Some students don't like writing and like thinking. They often sit there in a daze, repeating the memories they have learned.
……
Different students have different study habits. Developing good study habits suitable for your own situation will improve your study efficiency, and this study habit will naturally be maintained and will benefit you for life.
Mathematics learning has its own characteristics. For example, many people like to draw pictures when explaining mathematics, and always use the most intuitive and vivid language to explain the essential content; Some people always like to choose some familiar examples when explaining abstract mathematical concepts, and they will express abstract concepts clearly at once; Some people always give people a sense of wholeness when teaching mathematics. Naturally, source, process, result and application are indispensable. Express abstract concepts with intuitive images; Use concrete examples to understand general things; Continuously form an overall knowledge framework; Wait a minute. These are very good habits.
The formation of these good habits takes a long time to accumulate, and teachers are consciously or unconsciously influencing students with their own habits. I hope teachers can do this more consciously and actively. I also hope that students will become conscientious people, form some effective good habits suitable for their own conditions, change some bad habits and improve learning efficiency.
(3) Learn to "take"-active learning
From the teacher's point of view, I always hope to do everything possible to give my things to students. Some students don't know how to accept these things; Some students accept everything, good or bad; Some will pick and choose, the good ones will stay, and the important ones will be collected; Wait a minute. However, generally speaking, teachers like students who will take the initiative to "take".
We often say, "It is better to teach people to fish than to teach them to fish." Teachers generally think more about how to teach fish, which is very challenging. The "good study habits" mentioned above are the category of "catching fish".
"Teach them to fish" has two aspects, one is the method, and "good study habits" is the method; The other is motivation, such as curiosity, interest, ambition and understanding the value of mathematics. The two are inseparable, and "confidence" reflects the connection between them. It takes some effort to learn math well. When you encounter difficulties, you should stick to it. Some of our masters or doctors often encounter some obstacles when doing their papers. In addition to analyzing and discussing together, we have been asking for "persistence". This process can not only help them build confidence, but also "force" them to sum up "methods". Many excellent teachers are resourceful in this respect.
From the perspective of students, the main task of students is to learn, not only to learn "knowledge", but also to turn other people's knowledge into their own; We should also learn to "seek knowledge" and keep getting what we need. The two also complement each other. Need to think. For example, when doing problems, some students have a good habit. When they are finished, they should often think about this problem and make an evaluation of it. Is that a good question? What did you leave me? These ideas make their study "get twice the result with half the effort", which is how they seek knowledge.
We hope to combine "teaching and learning" and establish teacher-student interaction in this respect, which will be our glory. Teachers should try their best to provide students with more opportunities to improve their initiative, help students develop their potential, give different suggestions to non-students and let more students get started as soon as possible. Turn passivity into initiative.
(4) independent thinking and discussion.
Learning mathematics requires independent thinking. We need to think about the background, problems, concepts, theorems, applications and their relationships, so that they can naturally stay in our minds. We also need to do the problems and exercises independently, even if we ask others, and finally, we need to do them ourselves.
At present, various forms of seminars have become a basic working mode of learning mathematics. In the teaching of graduate students and some undergraduates, more and more seminars are used. The discussion forms are different, the levels are different and the number of people is different. But the basic form is the same, and there are clear discussion questions. Members attending the seminar should think carefully, prepare in advance, have special reports, and fully discuss and communicate.
This form can also be used for reference in primary and secondary schools. Teachers and students organize together, and everyone benefits.
With the help of the Internet, a number of special discussion platforms have emerged, especially some "famous teacher studios". It would be better if there were more discussions in this form. This is the greatest convenience brought by information technology, and we should make full use of it.