First, let students know what learning links are needed to learn mathematics well.
Before students begin to learn a certain mathematics textbook, our teacher must tell them that the following links should be paid attention to in learning mathematics well-the eight-link learning method:
(1) Make plans, (2) preview before class, (3) listen carefully, (4) review in time, (5) work independently, (6) solve problems, (7) summarize systematically and (8) study after class.
This method is summed up through the investigation of 200 excellent middle school students, 40 junior college students of Huazhong University of Science and Technology and 60 college students admitted to Wuhan University with high scores. As long as a student can learn according to these eight links and implement them step by step, he will become the master of learning and an excellent student in his class.
Each of the eight learning links also needs teachers' specific guidance, such as how to attend classes, how to preview, how to summarize and so on. In the first few weeks of each semester, the teacher should introduce it to the students step by step.
Second, let students clearly complete a math learning task, which needs to be completed step by step, so as to firmly grasp the knowledge. Because the process of mathematics learning is a complex cognitive process, to complete a mathematics learning task and truly master knowledge, all steps need to be completed. In psychology, the cognitive process is generally divided into four basic stages: perception, understanding, consolidation and application. In the four-round learning strategy, learning a lesson is also divided into four rounds. The first round: preview and find out the obstacles; The second round: class, breaking obstacles; The third round: review and remove obstacles; The fourth round: homework, apply what you have learned. In fact, these four rounds correspond to the perception, understanding, consolidation and application of the above cognitive processes. Although the angles are different, there are four stages, and the learning requirements of each step are very similar. Preview is to get a preliminary understanding of a lesson, lecture is to better understand the text, review is to consolidate, and homework is to apply what you have learned. Four-round learning strategy is a popular learning method in China in recent years. Because it conforms to the general cognitive process, if you study each lesson in strict accordance with these four steps, you will certainly achieve good results.
There are other ways to learn. According to different learning situations, the learning process is divided into four steps and five steps. Students can choose according to the characteristics of what they have learned, and even create their own learning steps, such as reading, listening, writing and practicing the four-word learning method, and then browse, ask questions, read, repeat and review the five-step learning method. Third, only by letting students know how to learn can they really master knowledge. If mathematical knowledge is regarded as a system, then the structure of mathematical knowledge has four elements, namely facts, reasons, uses and objects. Specifically, according to the different levels of knowledge structure, these four elements can be listed as follows:
Four things, facts, reasons, applications, materials, problems, topics, topics, topic methods, topic paths, questions, what, why, how to use, what inspiration, concepts, names, definitions, judgments, relationships, theorems, conditional conclusion proofs, applications, methods, formulas, expressions, deduction, calculation and connection.
In our opinion, no matter what level of knowledge we learn, we should master the corresponding four elements. If you only know "what" and don't know "why", you can't understand the principle of the conclusion. If you only know theoretical knowledge and don't know how to use it, it will become useless knowledge. If there is no clear thinking, the knowledge points are not closely linked and scattered, then the knowledge is not solid and the foundation is not solid, so it is difficult to innovate when learning new knowledge. Therefore, the four elements are indispensable. On the one hand, learners must walk in four steps and consciously master the four elements of each piece of knowledge. These four steps are: perception, understanding, application and systematization. Specifically, it is:
⑴ Perception (fact): Have a preliminary understanding of general conclusions and have an overall response to various attributes reflected by concepts, theorems and formulas. Perception is the beginning and foundation of mathematics learning. Only by knowing "what" can we further explore "why?" So as to understand and apply knowledge.
⑵ Understanding: In order to understand a mathematical conclusion, we must understand its principle and its context. Cognition is a kind of thinking process in which people gradually realize the various connections of things and find out their essential laws. It can be seen that only through understanding can the perceptual knowledge of things rise to rational knowledge. Connotation and extension of mathematical concepts, proof of theorems, derivation of formulas, explanation of conclusions, etc. Only by understanding can we really grasp the principle of mathematical facts.
⑶ Application: Application is the continuation and deepening of learning. On the basis of perception and understanding, students have mastered mathematical knowledge, but they should also apply knowledge to problem-solving and analysis, so as to deepen their understanding of what they have learned, make their study twice the result with half the effort, and master skills and improve their thinking ability through practical training. In mathematics textbooks, the summary of examples, the solution of exercises and the completion of extracurricular homework are all "practical" processes.
⑷ Systematization (event body): "event body" refers to "knowledge system". There are various connections between mathematics learning materials. If students understand the connection between old and new knowledge, they can do this and that. Mastering "things" has the following functions: knowledge structure is tight, memory is firm, thinking is flexible and diverse, which lays the foundation for learning new knowledge and is easy to produce new associations. Therefore, it is very important to systematize knowledge through summary.
Fourth, let students clearly learn a mathematical concept, theorem and formula from what aspects. In the process of learning mathematics, there are always a lot of concepts, theorems and formulas. How to really master them? Teachers should clearly point out what kind of process is needed, what requirements are to be met, and what aspects are generally to be understood and mastered.
Learning methods of mathematical concepts.
Mathematical concept is a form of thinking that reflects the essential attributes of mathematical objects. Its definition method is descriptive and denotative, and its ability of concept addition and classification is poor. A mathematical concept needs to remember the name, describe the essential attributes, realize the scope involved, and use the concept to make accurate judgments. These questions are not required by teachers. Without learning methods, it is difficult for students to study regularly.
Let's summarize the learning methods of mathematical concepts.
(1) Read the introduction and remember the names or symbols.
⑵ Recite the definition and master the characteristics.
⑶ Give two positive and negative examples to understand the scope of conceptual reflection.
(4) Practice and judge accurately.
⑤ Compare with other concepts and find out the relationship between them.
2. Learning methods of mathematical formulas.
The formula is abstract, and the letters in the formula represent infinite numbers in a certain range. Some students can master the formula in a short time, while others have to experience it repeatedly to jump out of the ever-changing digital relationship. Teachers should clearly tell students the steps needed in the process of learning formulas, so that students can master formulas quickly and smoothly.
The learning method of the mathematical formula we introduced is:
(1) Write a formula and remember the relationship between the letters in the formula.
⑵ Understand the cause and effect of the formula and master the derivation process.
⑶ Check the formula with numbers and experience the laws reflected by the formula in the process of concretization.
⑷ Transform the formula to understand its different forms.
⑤ Imagine the letters in the formula as an abstract framework, so that the formula can be used freely.
Learning methods of mathematical theorems.
A definite reason consists of two parts: conditions and conclusions. This theorem must be proved. Proving process is a bridge connecting conditions and conclusions, and learning theorem is to better apply it to solve various problems.
Let's summarize the learning methods of mathematical theorems:
(1) Recite Theorem.
⑵ Conditions and conclusions of the discriminant theorem.
(3) the proof process of understanding theorem
⑷ Applying theorems to prove related problems.
5] Understand the internal relations between theorems and related theorems and concepts.
Some theorems contain formulas, such as Vieta theorem, Pythagorean theorem and sine theorem, and their learning should be combined with formula learning methods.
Let students learn the method of self-study.
Self-study refers to a person's conscious activity of mastering knowledge, applying knowledge and acquiring skills independently without the help of others. Self-study is the best learning method in life, which mainly includes independent reading, independent thinking, self-organization, self-examination and self-supervision, and flexible use of knowledge to solve problems.
How can we effectively cultivate and develop students' mathematics self-study ability and form self-study ability? Wu Chuanhan put forward "self-study for ten times" in his "Learning Methods of Mathematics", that is, reading independently for a while and going in and out for a while; Three mistakes will win; Fourth, focus on energy; Five self-selected topics; Six will find their own materials; Seven will solve the problem; Eight will learn from others; Nine will make rational use of time; Ten will evaluate themselves.
There are several learning methods related to self-study, which are popular abroad. For example, fractional learning is an efficient and comprehensive learning method created by American scholars, which is popular all over the world. The specific steps are: browsing, copying titles, setting goals, reading and evaluating. Another similar learning method, also created by Americans, is called SQL2R learning method. The specific steps are browsing, asking questions, reciting and reviewing. Both methods can achieve good results in self-study.
In the process of self-learning mathematics textbooks, according to the characteristics of mathematics, we respectively put forward two methods: algebra self-learning and geometry self-learning:
Algebraic learning method.
(1) Copy the title, browse and set the target.
⑵ Read and record the key contents.
(3) Try to make an example. Do exercises quickly and summarize the questions.
5] Memory summary.
Four steps of geometry learning.
(1).( 1) Write the topic and browse the teaching materials; (2) Teach yourself and write the contents;
⑵. ① Read the teaching materials according to the catalogue, ② Teach yourself geometric concepts and theorems;
(3).① Reading examples to form ideas, ② Writing out the process of solving problems;
(4).① Do the problem quickly, ② Summarize the method of solving the problem.
Let students understand how to achieve efficient learning results. Super learning method, fast memory method, fast reading method and fast learning method are all learning methods with the purpose of "fast", so they can be called efficient learning method. The core of learning mathematics knowledge is memory, and being able to remember what you have learned quickly and firmly is the main symbol of good study and good grades. Therefore, exploring and mastering various quick memory methods should be the goal that every student must pay attention to summarize and pursue. Many books are introducing the fast memory method, the central idea is novelty and excitement, turning words into images (because the memory of images is 1000 times that of words), relaxing the brain and enriching associations. Memory is closely related to attention, observation, imagination and understanding. The cultivation of memory must be coordinated with other intellectual factors. Psychologists divide memory into four steps: memory, retention, recognition and reproduction, and we should also remember mathematical knowledge according to these four steps.
Third, the cultivation of talented students in mathematics.
In mathematics learning, students who study hard and get excellent grades are usually called "top students". Comparatively speaking, gifted students have outstanding academic achievements and active thinking in class; The task assigned by the teacher can be easily completed by taking the lead. As their math teachers, they should have two clear goals for the cultivation of talented students. First, we should actively guide and strictly demand, so that their strong thirst for knowledge can be satisfied, so that they can make more use of their studies and their mathematical talents can be fully demonstrated. Secondly, they should influence and help other students to develop together through the positive attitude of gifted students and better learning methods. We can regard them as "little teachers" (we can also call them mentors) to help teachers carry out their work, which not only cultivates top students' deeper understanding and systematic mastery of knowledge, but also makes the class's academic performance improve continuously.
The general work of cultivating talented students in mathematics.
Carry out extracurricular activities in mathematics to broaden the horizons of gifted students. Eugenics in mechanics. With the basic knowledge mastered, we can carry out rich extracurricular activities for them, such as answering interesting math questions: reading extracurricular books about math, writing monographs on learning math, telling stories about math and mathematicians, summing up mathematical thinking methods, and solving practical problems within our power. In addition, teachers can provide more exercise opportunities for gifted students through activities such as math lectures or mathematicians' reports, math lectures and math competitions.
Active guidance in the classroom can make gifted students step into a higher and broader thinking space. A student in Tsinghua recalled that he had been a monitor in high school for three years, and his study was particularly excellent. But most math classes think the teacher is too shallow, because he has the ability to understand all the problems by reading the textbook once. So in order to be a good student who abides by discipline, he listened to the three-year class he has always understood. It can be seen that the cultivation and guidance of gifted students in the classroom, like poor students, has really become a corner forgotten by teachers. Therefore, in the classroom, teachers should start with poor students and end with gifted students, with simple examples and thought-provoking questions, and appropriately supplement the examples and exercises in the classroom to put forward higher requirements for gifted students. Let eugenics break away from the teacher's lecture process and think about more interesting and difficult problems.
Guidance on learning methods of gifted students in mathematics.
Although the learning methods of gifted students may be better for poor students, there are still many problems. In order to make them get better grades, teachers should put forward better learning methods and let talented students work hard to achieve them.
1. Excellent students should strictly implement the "eight-link learning method" and constantly improve their learning enthusiasm and initiative. Many excellent students consciously demand themselves in an all-round way according to the "eight-link learning method", but only a few top students in each class can strictly achieve all links. Teachers should always remind them not to be complacent, to study hard and to be proactive.
2. Let talented students master some demanding learning methods suitable for their continued study. Some learning methods have higher requirements and poor operability, which are not suitable for poor students, but are of great help to excellent students. For example, Paulia's problem-solving thinking method is divided into the following five steps: (1) foresight method; (2) collecting and sorting data; (3) Identify and recall, enrich and rearrange; (4) separation and combination; 5] Review. In addition, Paulia also mentioned the problem-solving methods in the book: ① find out the problem; (2) ask questions; ③ Implementation plan; 4 review. These two methods can only be used when gifted students reach a certain level of thinking development in the process of solving mathematical problems.
For another example, if the eugenics have mastered the basic knowledge and methods of geometric proof, and have been able to express the proof process smoothly and accurately, then they need to change their learning methods, and their learning goal has become to accumulate proof ideas and problem-solving skills of various geometric problems, then we can tell the eugenics a reduction method to improve their geometric proof ability:
(1) Review the questions, find out the known conditions and verify the conclusions.
⑵ Draw a picture as an auxiliary line to find a way to prove the problem.
⑶ Record the key steps to prove the problem.
⑷ Summarize the thinking of syndrome, so that the process of syndrome can form a clear impression in the brain.
Reduction is a method to classify knowledge as known. When we encounter a new geometric proof problem, we need to pay attention to its type and find the key steps. Every key step includes familiar little details, which we will omit and end when it is classified as a known type. At this time, the most important thing is to remember the steps of transformation and the idea of proving the problem, and no longer pay attention to the detailed expression process.
Let gifted students have a high-level understanding and mastery of the basic thinking methods of mathematics learning. There are six pairs of basic thinking methods in mathematics learning 12: observation and experiment; Analysis and synthesis; Abstraction and generalization; Comparison and classification; General rest and specialization; Analogy and induction. In the process of mathematics learning, we should add three pairs of thinking methods: transformation and flexibility. Transition and contraction; Organization and systematization. In the process of solving mathematical problems, it is necessary for a gifted student to be able to apply these thinking methods freely.
Third, advocate gifted students to be small teachers, and exercise their thinking while helping poor students learn. Since gifted students are excellent in all aspects, we can introduce their excellent learning experience and good learning methods to other students through their individual counseling for poor students. We can divide the class into more than ten groups, and each group is headed by a talented student. This group leader is called a mentor. A tutor is a learning leader selected from the students. He is both a student and a little teacher to other students. He should take the lead in learning and help other students make progress together. The tutor is also the first "rich" person in our educational reform. In class, they first understand how to study under the guidance of the teacher. Know how to read, learn by yourself, listen to lectures, summarize, preview and learn actively. Then, they pass on this learning experience to other students, and finally achieve the goal of making progress with the whole class. Many outstanding problems in the class teaching system can be solved by using tutors to carry out counseling, evaluation, discussion and mutual learning. In addition, tutors have also been trained in these activities, because being able to explain a problem smoothly can greatly deepen their impression, many vague questions are organized and clarified, and simple questions are understood more deeply. This is the basic principle of fast learning method.
4. Students with poor mathematics.
Because the students in any class will be divided into three levels with the progress of teaching, some students find it difficult to learn mathematics and can't keep up with their work. Therefore, organizing students with learning difficulties to participate in teachers' purposeful activities is an effective way to improve the quality of mathematics teaching in a large area.
To transform students with learning difficulties, teachers should do a good job in ideological education and academic counseling for all kinds of students in accordance with the principle of teaching students in accordance with their aptitude, so that they can all improve and develop themselves. Generally speaking, there are many reasons for students' poor grades. First of all, their intelligence development level is low, and their observation and analysis ability is poor. Secondly, their non-intelligence factors are also poor, their curiosity is low, they lack confidence in learning, their attitude towards mathematics learning is incorrect and they are not interested. To do a good job in spreading the learning of students with learning difficulties, teachers must deeply understand the reasons why students with learning difficulties are backward, strive to develop their intellectual and non-intellectual factors according to the actual situation of students with learning difficulties, introduce suitable learning methods in a planned way, and make a series of learning method guidance from each learning link.
1. Give priority to cultivating non-intellectual factors of students with learning difficulties. Non-intelligence factors play a dynamic role in the learning process. Many poor students often show a lack of interest in learning mathematics and a strong will to overcome difficulties. To solve this problem, in addition to teachers' constant concern for getting close to them and guiding and encouraging them, they should actually be introduced to some methods to cultivate their interest and exercise their will, and provide some activities that they can enjoy learning.
1. The operation mode for students with learning difficulties to consciously cultivate their interest in mathematics learning. Look at some interesting math materials. Consciously appreciate the beautiful characteristics of simplicity, unity, symmetry and ingenuity in mathematics. (3) to find and solve math problems directly related to oneself. ④ Learn mathematics in the game. ⑤ Determine the small goals of learning and experience the joy of success. 6. Solve problems and read books with your favorite friends. When you can't understand the textbook, try to copy it and concentrate on your study slowly. ⑦ If you understand a lesson, you will solve a problem and gradually become interested in mathematics.
2. Operation mode of exercising strong will quality. Mathematics learning is more difficult than other subjects, which requires hard work and stronger perseverance and patience. Students with learning difficulties are often determined to study hard, and will soon be replaced by various desires, so that their learning mind can not concentrate on learning. Therefore, we give the following suggestions to exercise our will: ① Write down our vows in front of us, set a goal, have the idea of not giving up until we reach our goal, and praise ourselves once and for all, thus gradually extending the study time. The way to change your mind suddenly, when a non-learning activity attracts you very much, suddenly tell yourself to study, so as to overcome your original desire and succeed, which shows that you have become a strong-willed "giant". ③ Use the characteristics of mathematics itself to cultivate one's consciousness, tenacity and self-control. ④ Learn to stick to the plan, finish math homework on time, and form the habit of self-inspection, self-supervision and self-encouragement.
2. Developing intelligence factors is an urgent task for students with learning difficulties. Lack of concentration, poor memory and poor imagination make it difficult for students with learning difficulties to learn mathematics at the same price as those with good grades. Recite the same mathematical law once or twice, and gifted students may not be able to recite it more than ten times. Every time this happens, students with learning difficulties will think that they are "born stupid" and give up learning. At this point, teachers should make it clear that memory requires methods and memory ability is also acquired through exercise. The super learning method and quick memory method mentioned above should be introduced to poor students for trial use. Calm yourself down by taking a deep breath and reach the alpha wave state by self-adjustment. This programmed training may make a "stupid child" smart immediately. As long as students with learning difficulties can quickly remember the contents of the textbooks they have learned, their learning situation will soon change.
For students with learning difficulties, there should also be specific guidance on learning methods.
It is common to see some children recite texts and formulas, just reading and writing repeatedly, but not analyzing and thinking, reviewing and telling themselves, while eugenics uses a completely different and efficient method, that is, remembering and repeating when it begins to recite. Thus, for students with learning difficulties, some very small learning links still need guidance. What should I do if I can't solve a math homework problem? This is a common problem for gifted students, but students with learning difficulties can't imitate writing by reviewing textbooks and reading examples, so as to understand the problems they face. Mentality, mental outlook and the combination of listening, writing, reading and speaking are the key factors that affect the effect of listening. However, students with learning difficulties have been listening to math classes inefficiently for several years. This specific operation method is the main reason for the poor learning effect of students with learning difficulties and needs the guidance of teachers.
Some specific methods are also introduced in the four-round learning method, such as the four-round review method: ① Read through and review systematically; 2 intensive reading, focusing on review; (3) Exercises, reviewing and solving problems; (4) Recall and test review. Four-step problem-solving method: ① Examine the problem and find out what it is; 2 conceive and find out the reasons; 3 answer, think clearly what to do; (4) Inspection and verification. Four-step memory method: memory, retention, recognition and reproduction. These seemingly ordinary steps, but once you can follow them, the learning effect will immediately appear.
Some students feel at a loss when solving math problems and don't know how to think. Then we can introduce Paulia's self questioning method in solving mathematical problems, so that he can learn to think and explore.
(1) What kind of solution did I choose?
Why did I make such a choice?
(3) What stage have I reached now?
(4) What is the position of this step in the whole problem-solving process?
5] What are the main difficulties I am facing at present?
[6] What is the prospect of solving the problem?
In different stages of mathematics learning, the learning methods should be changed accordingly, which also requires the careful guidance of teachers. For example, in the second semester of senior one, students always find it difficult to get started when studying solid geometry in senior two and senior one. Many old teachers agree with the following methods. Poor students can try it in class or by themselves.
1. Look at the topic drawing (or copy the topic drawing);
2. Examine the questions to find ideas (or listen to the teacher explain ideas);
3. Read the proof process in the book;
4. Recall and write the proof process.
How to preview, how to attend classes, how to review in time and how to summarize are all unknown to students with learning difficulties. Teachers can refer to the "eight-link learning method" and give guidance according to the actual situation of students.
Finally, let's introduce three basic principles of Noguchi's super learning method: ① learning interesting things; ② Starting from understanding the whole content; 3 understand that 80% is moving forward. It is very important for students with learning difficulties to abide by these three principles. Because you have to learn what you are interested in first, so it is easy to get started. After you have some experience, you will have new interests. To understand the whole content, you need to remember the general framework and grasp the key content. Don't want to remember everywhere. The mental burden is too heavy, which easily makes students with learning difficulties lose confidence. 80% of them have mastered the general content, and some people will understand it automatically after learning the following content. Therefore, students with learning difficulties need not worry about problems they don't understand, but should learn new knowledge with confidence.