1 how to cultivate mathematical logical thinking ability
Strengthen training and cultivate the flexibility of students' thinking
In order to keep students' memory of knowledge and develop their flexible thinking, teachers should strengthen students' topic training and improve their problem-solving ability. In problem-solving teaching, we should pay attention to the training of various types of questions. Self-made questions should not only consider the rationality of structure, the logic and rigor of quantitative relationship, but also consider the flexibility of thinking. The process of compiling questions is actually the process of cultivating students' initial logical thinking. The practice of multiple solutions to one question not only cultivates the flexibility and creativity of students' thinking, but also stimulates their initiative and enthusiasm in learning. In order to enhance the flexibility of mathematics teaching, teachers can also encourage students to cooperate to solve problems. Because of the characteristics of mathematics, a mathematics discipline can have many ways to solve a problem. In view of this feature, cooperative inquiry learning can be used to teach the process of mathematical problem solving in the teaching process.
Divide students into groups, take problems as the fundamental factor to drive teaching, and follow the steps of "cooperative preview, exploring answers, inspiring and guiding, and consolidating and expanding". First of all, teachers ask questions according to the syllabus, and students design and exchange views on the problems in groups. Then let the students solve the problem interactively, find the answer to the question through various channels, and broaden the students' thinking. In the process of solving problems, teachers can inspire and guide students to solve problems and explain common problems. Finally, through the publicity and explanation of answers and problem-solving ideas in each group, all students can understand different problem-solving ideas. Teachers then evaluate students' knowledge, systematize students' mastery of basic knowledge, combine students' educational reality or social hot issues, sublimate students' thinking, and apply what they have learned. In the teaching process, students' logical thinking ability is fully emphasized, so that students can learn to think in learning, which not only cultivates the flexibility and creativity of students' thinking, but also stimulates their initiative and enthusiasm in learning.
Clarify concepts and establish the integrity of students' thinking
Mathematical concepts are abstract, rigorous and systematic, and the psychological characteristics of primary school students are easy to understand and accept concrete and intuitive perceptual knowledge. Therefore, at the beginning of teaching, it is necessary to build a bridge between mathematics and life, provide rich typical and comprehensive perceptual materials, and do everything possible to enrich students' perceptual materials. There are various ways to introduce concepts, which can be introduced intuitively or from situations and students' lives. When designing specific situations, teachers should not cut to the chase and cover everything, but should introduce them step by step according to the age characteristics of primary school students, closely linked with the existing knowledge and experience of students. At the same time, we should also pay attention to the fact that the introduction situation of concepts should highlight the essential characteristics of concepts, and the situation must be related to the essential attributes of concepts, otherwise it will affect the teaching effect because it is far away from the teaching content, and sometimes even lead students to go astray.
The introduction path should reflect the background of the concept. Teachers should teach students in accordance with their aptitude, choose the best introduction path, try to eliminate the interference of non-essential attributes, let students touch the essential characteristics of concepts as soon as possible, and reflect the efficiency of the concept establishment process. Mastering concepts is a complex cognitive process. Primary school students' mastery of concepts is often not completed at one time, but from concrete to abstract, and then from abstract to concrete repeatedly. When students initially establish concepts, they need to use various methods to promote the maintenance of concepts in students' cognitive structure, deepen their understanding and memory of concepts through continuous use, and consolidate the newly established concepts. Concepts are always taught one by one, so in the eyes of primary school students, concepts are often isolated. When teaching reaches a certain level, students should be guided to put the concepts they have learned together, find the vertical or horizontal relationship between concepts, and form a concept system, so that the mathematical knowledge in textbooks can be transformed into the cognitive structure in students' minds, which is conducive to students' retrieval, extraction and application of knowledge, promoting the transfer of knowledge, establishing the integrity of students' thinking and developing their mathematical thinking ability.
2 Mathematical thinking training
Emphasize participation and innovation.
The new curriculum standard puts forward to cultivate students' inquiry ability, and the content of mathematics classroom teaching is extrapolated. Teachers should change their ideas and establish a new teaching concept. Mathematics is not only the knowledge in the ivory tower, but also a practical subject. It is necessary to create colorful mathematics learning situations, typify mathematics problems in life, make mathematics problems live, let students participate in mathematics practice unconsciously, narrow the distance between students and mathematics, touch students' desire to discover, study and solve problems, and thus generate interest in learning mathematics. Under the guidance of teachers, students actively participate in creative development. The leading role of teachers lies in how to make students become the main body of development. In the math class, we should give students full opportunities to participate independently, have a good democratic atmosphere, encourage more and criticize less, build up students' confidence, and use teaching materials to let students ask their own math questions according to the situation. Teachers guide students to ask reasonable questions in time, stimulate students' interest, and draw conclusions by students who can operate. In this way, students' thinking has been developed in a subtle way, rather than being imposed on them by teachers. Of course, teachers should also pay attention to the mistakes found in students' exploration, analyze the causes of the mistakes and guide students to develop in the right direction.
In this way, our previous research on teaching methods will be transformed into the research on learning methods. Only when students learn to learn, can they innovate in their studies and show their individuality. From a mathematical point of view, there is only one correct answer to things. Where should innovation start? All roads lead to Rome, and there is only one goal, but there can be multiple roads leading to the goal. The answer in mathematics is often yes, but there are many ways to solve the problem and find the answer. In teaching activities, teachers should play the role of guides, help students to study different problem-solving methods, highlight the thinking of seeking differences, encourage students to make bold assumptions, and verify them with students carefully. Don't completely pursue the perfection of the answer, the key lies in the process of students' exploration and thinking. Students can learn actively in the learning situation to enrich the process as much as possible, even if they get the wrong answer, which is a very practical mathematics learning practice.
Emphasize thinking and cooperation.
The author thinks that thinking is the core of intelligence, and we should attach importance to students' thinking process of acquiring knowledge. From the point of view of thinking, the criticized sea tactics are nothing more than repeating the thinking process of solving problems, so that thinking can be internalized into its own way of thinking in repetition, thus forming the ability to solve problems. Fundamentally speaking, it is to train students' thinking and pay attention to the formation process of students' thinking. It's just that this method is too mechanized and formal. Also known as the "sea", it is obviously biased, and it is too late. It is necessary to guide students to make abstract generalization, simple judgment and reasoning, comparison, analysis and synthesis on the basis of perceptual materials through operation and observation, so as to cultivate students' preliminary logical thinking ability. Cultivating students' thinking ability should run through the whole process of classroom teaching.
For example, when you talk about the division problem in one step, students say the formula first, and then talk about your ideas. Let the number of copies and each copy be calculated by division, which has an abstract impression in students' minds. Therefore, we can further grasp that one number is several times that of another number, which is derived from the number of copies, and we can draw inferences from others. Pay attention to the actual process of students' thinking, and see if students have thought about problems and how many ways to think. Communication and cooperation often lead to inventions. Therefore, in the teaching process, we should pay attention to cultivating students' cooperative spirit and fully embody the multi-directional communication between students and between teachers and students. Although we advocate cooperation, students must be allowed to think independently before cooperation, so that cooperation and communication can have a purpose. Through discussion among students, we can share resources and cultivate the spirit of cooperation.
3 Mathematical thinking training
Focus on cultivating students' analytical ability and comprehensive ability.
Analysis and synthesis are the basic process of thinking and an important logical thinking method. According to the characteristics of students, when teaching practical problems, my usual practice is to guide students from analyzing and synthesizing with the help of line graphs to analyzing and synthesizing according to given conditions and problems, and attach importance to concept teaching, calculation teaching and geometry preparation knowledge teaching, so as to cultivate students' analytical and comprehensive ability.
For example, after studying cuboids and cubes, I showed such a question: "A cube block with a side length of 8 cm is completely painted with red, and then it is divided into small cubes with a side length of 2 cm, of which three sides are red, two sides are red and one side is red. How many pieces are there without red?" At first glance, this problem seems difficult to start with. First of all, I am not in a hurry for students to calculate. Instead, I want students to tell the characteristics of the cube first, and then let students discuss how to divide the big cube into small cubes of 2 cm. After reaching a unanimous conclusion, let them think: How many small cubes are there? Think again: where are the small pieces of wood painted red on three sides, two sides and one side of the big cube before cutting? (Drawing can help analysis) After figuring out these questions, I guide the students to answer according to the situation. Through analysis, the students deduce the answers.
Pay attention to the cultivation of students' abstract generalization ability and reasoning ability
First of all, I asked such a question: "It takes 15 hours for Xiao Wang to process 900 parts by himself, and 15 hours for Xiao Li. How many hours does it take for two people?" After the students analyzed the quantitative relationship and asked for answers, I put forward two questions for students to answer: 1. When processing 1800 parts, it takes 10 hour for Xiao Wang to do it alone, 15 hour for Xiao Li to do it alone. How many hours does it take for two people to do it together? 2. Processing 180 parts takes 10 hour for Xiao Wang and 15 hour for Xiao Li. How many hours does it take for two people to do it together?
After answering, I asked the following questions: (1) If we continue to change only the total number of parts to be processed, will it change the time for two people to complete the task together? how much is it? (2) Why only change the specific total workload without changing the cooperation time? (3) Can "a batch of parts" be used instead of the specific quantity? (4) The total workload is expressed in "1". What application problem is this? (5) What quantities have been studied in this fractional application problem? After the answer, the teacher told the students in a positive tone that such a question is called the fractional application problem of learning engineering problems. The development from the work problem of integers to the engineering problem of fractions is an abstraction of the essence of knowledge and a leap in solving problems. In the whole teaching process, students use their existing knowledge to think about problems and draw their own conclusions through logical thinking activities such as comparison, analysis, abstraction and generalization. They not only master knowledge on the basis of understanding, but also develop the ability of abstract generalization and reasoning in the process of seeking knowledge.
4 Mathematical thinking training
Guide active migration and promote the process of transforming old knowledge into new knowledge.
The process of mathematics teaching is that students systematically learn the indirect knowledge of predecessors under the guidance of teachers, guide students to actively transfer knowledge, and promote the transformation of old knowledge into new knowledge, which is a shortcut for students to inherit the experience of predecessors. There are * * * the same factors among all parts of primary school mathematics textbooks, so they are organically related: explore this factor, communicate its connection, guide students to transfer the known to the unknown, assimilate new knowledge into old knowledge, let students reason with the obtained judgment, and then obtain new judgments, thus expanding the cognitive structure.
Therefore, on the one hand, when teaching new knowledge, we should pay attention to arousing the old knowledge that has been learned. For example, when teaching division in which divisor is decimal, it is necessary to arouse the reappearance of old knowledge such as "the nature of quotient invariance" and "the law of decimal size change caused by decimal position movement"; On the other hand, we should pave the way for new analogical knowledge. For example, to help students understand the meaning of multiplying a number by a fraction, it is necessary to help students understand that multiplying a number by an integer by a decimal is … so that the knowledge that students have mastered in previous studies can become "the internal stimulus and driving force for establishing new connections".
Strengthen practice guidance and promote the application from general to individual.
In order to understand the concepts, principles and methods, students should not only go through the development process from the individual to the general, but also go through the regression from the general to the individual, that is, apply the general law to solve individual problems, which is a process of knowledge concretization accompanied by the thinking process. Therefore, first, we should strengthen basic exercises and pay attention to the understanding of basic principles;
The second is to strengthen variant practice, so that students can realize the concretization of knowledge in different mathematical artistic conception, and then get a more generalized and generalized understanding; Third, we should pay attention to the comparison in practice, so that students can get a more specific and accurate understanding; Fourth, strengthen practical exercise and promote students' "action thinking".
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