1 how to cultivate students' mathematical and logical thinking
(1) flexible thinking. The flexibility of thinking shows that the thinking subject can flexibly adjust the original thinking mode based on the existing experience according to the changes of the thinking object, so that the new thinking can solve the problem more efficiently. For primary school mathematics, the flexibility of thinking is very important, and the method of solving mathematics is not important. In the process of solving problems, students can change their problem-solving methods and ideas according to different types of questions, so as to find a more suitable problem-solving method, which is mainly manifested in the problem-solving methods such as multiple solutions to one problem, variable exercises and deformation of the same solution. For example, 200 kilograms of seawater can make 2.5 kilograms of salt, so how many kilograms of salt can 50 thousand kilograms of seawater make? This is a multi-solution problem, which can be solved by 2.5 ÷ 200× 50000; 50000÷(200÷2.5); 2.5×(50000÷200) several methods to solve.
(2) Deep thinking. The profundity of thinking is the ability to see the essence through phenomena, which is the basis of thinking quality. In primary school mathematics, the main performance is that superficial phenomena can lead to in-depth thinking, thus discovering the internal laws and internal relations of problems and finding solutions to problems. Teachers can train their thinking through open exercises.
(3) The thinking is original. The originality of thinking means that thinking has the level of independent creation. Therefore, teachers should encourage students to imagine boldly, find a variety of problem-solving methods, and find out the simplest method, not limited to the conventional problem-solving mode. For example, if you group 2.5.6 digital cards, according to the conventional thinking mode, the number will only be 25.26.256.265.52.56. In addition to these numbers, students can also find the characteristics of "6", and using "6" as "9" in turn will also form more numbers.
(4) Thinking is the key. Critical thinking means that the thinking subject can find his own mistakes in the process of thinking and consciously correct them through independent thinking, with the ability to dare to question and strong discrimination. In the teaching process, teachers should actively guide students to think independently and be good at finding their own problems in thinking, so as to solve problems independently. They should guide students to think from different angles, test and reason their own conclusions, and explore new ways to solve problems. Students should also be encouraged to question and ask questions. The process of asking questions is also a process of thinking, which is conducive to the cultivation of students' critical thinking.
2 How to cultivate students' mathematical thinking
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.
Re-cultivation for development
Mathematics classroom teaching should take development as the classroom teaching goal. We teachers can't just pay attention to the immediate interests and complete the knowledge goals, but ignore the cultivation of students' ability and the improvement of other aspects of quality. In mathematics teaching, we should not only impart knowledge, but also cultivate ability. We should put intellectual development and ideological education throughout the teaching, pay attention to the combination of knowledge and ability, process and method, emotional attitude and values, and fully embody the unity of ideological education, imparting knowledge and cultivating ability. Mathematics teachers must have a developmental vision.
For example, there are picture application problems in the first grade of primary school, one is subtraction application problem, and the other is addition application problem. As far as subtraction application problems are concerned, if students use addition to calculate according to their existing knowledge and experience, our teacher should not deny it, because such application problems are exactly the same as the equations of senior grades. If you deny it easily, it will undoubtedly dampen the enthusiasm of children to explore. It is necessary to guide students appropriately, start with life experience, affirm the method of homeopathic thinking, and guide them to do subtraction thinking at the same time. Why are you doing this? From the long-term goal, from the perspective of mathematical thinking, students only have a good foundation, and teachers are not easy to blindly kill students' subconscious equation solutions. That is to say, teachers should look at the mathematical system of primary schools with a developmental perspective and treat students. At the same time, the knowledge and experience of first-year students often come from life, which is a typical image thinking and should be protected from the perspective of development. No matter vertically or horizontally, we should have a developmental vision and focus on students' lifelong development. Only by climbing high can you see far.
3 How to cultivate students' mathematical logical thinking
Methods of analysis and synthesis
The so-called analytical method is to decompose the research object into its various components, and then study each component separately, so as to obtain a way of thinking to understand the essence of the research object. The comprehensive method is to study all parts of the cognitive object as a whole and understand its essence. For example, if the students know 5, the teacher asks the students to put 5 apples on two plates, thus getting four points: 1 and 4; 2 and 3; 3 and 2; 4 and 1. From this, students realize that 5 can be divided by 1 and 4, or by 2 and 3. This is the analysis method. On the other hand, teachers guide students to understand on the basis of analysis: 1 and 4 can form 5, 2 and 3 or 5. This is a comprehensive approach. On this basis, teachers can once again use the method of analysis and synthesis to guide students to understand that 5 can be divided by 5 1, so as to know that there is 51in 5; Conversely, five 1 can make up five. Analysis and synthesis are widely used in integer recognition, fractions, decimals, elementary arithmetic, compound application problems, combined graphic calculation and other teaching.
Method of abstract generalization
Abstraction is a way of thinking that abandons individual and non-essential attributes from many objective things and extracts common and essential attributes. Generalization is to integrate the common essential attributes of similar things into a whole. For example, there are 45 addition questions within 10. When students begin to learn, they all calculate by memorizing the composition of numbers. However, if the teacher helps the students to gradually and abstractly summarize the following laws, the students' calculation will be much more flexible: 1. Add 1 to a number and the result is the successor of this number. 2. Exchangeability of application addition. 3. A number plus 2, a total of 13, can be deduced by the law. 4.5+5= 10。 By mastering these laws, students can reduce their memory burden and greatly improve their understanding level. For another example, when the calculated number is the addition of 1 1, students can calculate several problems such as 2+9, 3+8, 7+4, 6+5 by swinging the stick, and then abstract the "ten-complement method" from them: look at large numbers, divide decimals, then add a few. In this way, when you learn all the carry addition within 20, you can directly use the "complement ten method" to calculate. Facts show that once students master the abstract learning method, mechanical memory will be replaced by meaning understanding, and cognitive ability and thinking ability will make a new leap.
Methods of comparison and classification
Comparison is a method to determine the similarities and differences between research objects and phenomena. Only comparison can distinguish, which is the basis of human thinking. Classification is the basic method to sort out the facts of processing science. Comparison and classification run through the whole process of primary school mathematics teaching. For example, when students begin to learn mathematics, they will compare the length and size, and then compare how much they learn. Then the same size will be put together and the same shape will be grouped together. Or mathematics that combines the same attributes (integer, decimal, fraction). The former embodies the comparative method, while the latter illustrates the classification method with examples. Classification is often obtained by comparison. Comparative method and classification method are the most basic thinking methods often used in primary school mathematics teaching.
4. Cultivate students' mathematical logical thinking ability.
Cultivation of analytical and comprehensive ability
Analysis is the decomposition of things or objects into various parts or attributes; Synthesis is to combine various parts or attributes of things or objects into a whole. Analysis and synthesis are two closely related logical methods. Things can't be synthesized without analysis, so it runs through people's whole cognitive activities and plays an equally important role. Analysis and synthesis are widely used in primary school mathematics teaching.
For example, in the fourth volume of the mathematics textbook for six-year primary schools in compulsory education, when teaching the oral calculation of dividing one digit by two digits, students are guided to divide 69 into six tens and nine ones, and six tens are divided into three parts, each of which is two tens and nine ones are divided into three parts, each of which is three ones, and the sum of two tens and three is 23, which is the required quotient. On this basis, show the complete steps of oral calculation. In the teaching of written arithmetic, we introduce oral arithmetic in which one digit is divided by two digits. Let the students talk about what they think and how much they get when they do oral calculation, and then explain to the students that division can also be calculated vertically, so as to list a division that is calculated vertically. This kind of teaching not only helps students to understand the arithmetic of oral calculation and written calculation, but also helps to cultivate students' preliminary analysis and comprehensive ability.
Cultivation of abstract generalization ability
Abstraction is the process of putting forward the common and essential attributes between various objects or phenomena and separating the non-essential attributes. Generalization is the process of unifying the common and essential attributes of abstract things. In mathematics, any number, formula, concept, nature and other knowledge is the result of abstraction and generalization. Abstract generalization must be based on a large number of perceptual materials, without which there is no basis for understanding. Pupils especially need to use intuition to form perceptual knowledge, but intuition is only a means to arouse students' positive thinking, not the ultimate goal.
After students get rich perceptual materials and form representations, they should make abstract generalizations in time, reveal the essence or laws, and make their cognition reach a rational stage. For example, in the fourth volume of the mathematics textbook for six-year primary schools in compulsory education, the teaching multiplier is a one-digit multiplication. When the unit product reaches 65,438+00 and needs to be rounded up, first multiply one digit by two digits, and let the students set up sticks. Then show a stick diagram, which shows that three four sticks are 65,438+02, of which 65,438+00 are tied into a bundle and placed under three rows of sticks. Then abstract the multiplication vertical form against the bar chart. After the trial, an example of multiplying one digit by three digits or four digits appears, which shows that the product of ten digits and hundred digits should be rounded to hundreds and thousands when it reaches 10. Finally, on the basis of learning three examples, guide students to summarize the multiplication rule of multiplier as single digit, and cultivate students' abstract generalization ability at the same time.