1 How Mathematics Infiltrates Logical Thinking
Gradually cultivate students' abstract thinking ability
Compared with junior high school mathematics, the most important feature of primary school mathematics is that students can find concrete things to help them think in the process of thinking, which is also an effective learning method of mathematics introduction, which can effectively speed up students' mastery and deepen their understanding in the early stage of mathematics learning. However, after entering junior high school, the appearance of geometric figures and algebraic expressions requires students to abandon auxiliary tools and think abstractly. Some students change slowly, which leads to a decline in their grades and a blow to their self-confidence. Therefore, in practical teaching activities, teachers should make more efforts on the guidance of abstract thinking, familiarize students with the significance and practical application of algebra, and cultivate students' abstract thinking ability in solving exercises.
For example, when proving that triangles are congruent, many students do not solve problems according to the conditions and theorems required by the topic, but subjectively "look", first to see whether two triangles are congruent, and then to prove. Over time, students' abstract thinking ability gradually decreases, which can't lay a good foundation for studying solid geometry in the future. At this time, teachers should actively guide students to recall the congruent triangles proof methods they have learned, such as "angle proof method", and gradually get rid of the habit of "seeing with eyes" by applying theorems.
Through comparison and contrast, strengthen students' ability of contact and discrimination.
Comparison in mathematics refers to comparing the characteristics of two or more research objects. Contrast is the basis of understanding and thinking. With the increasing amount of junior high school students' learning knowledge, mastering the similarities and differences between knowledge points has become an important way to consolidate students' learning. For example, in the teaching of positive numbers and negative numbers, teachers can guide students to realize that positive numbers are relative to negative numbers, and negative numbers would not exist without positive numbers. Above 5 meters above sea level, it is marked as "+5", and below 5 meters above sea level, it is marked as "-5". Through comparison, students can easily grasp the similarities and differences and form correct mathematical concepts.
There are many confusing rules, concepts and laws in junior high school mathematics. Through intuitive comparison, students' logical thinking ability can be effectively strengthened and students can master what they have learned. For example, when learning "one-dimensional linear inequality", do problems and solve 2 (x+2) >; 3(3x-4)+5 and 2(x+2)=3(3x-4)+5. If the teacher compares the solutions of the two problems, students can easily understand that the first steps of the two problems are the same, but there are differences when the coefficient is changed to 1. Through this contrast, students have formed a strong impression on the differences and mastered the knowledge they have learned more deeply.
2. Cultivate students' mathematical thinking methods
Adapt to the age characteristics of students' thinking development and attach importance to the thinking process.
Pupils are in the stage of gradual transition from concrete image thinking to abstract logical thinking. Students of different ages have different thinking characteristics. In order to achieve good results, we should cultivate students' thinking ability consciously and systematically according to the characteristics of students' thinking development in teaching. For example, junior students are young, have little life experience, have the advantage of concrete thinking in images, and their abstract thinking ability is still very weak, so they often can't distinguish the essential characteristics of things, and often can't tell what they think when solving application problems, or can't fully express their own problem-solving ideas. When teaching, we should combine operation with intuition, ask enlightening questions, guide students to analyze and compare step by step, and find out regular knowledge or methods to solve problems. Students sometimes express incorrectly, so teachers should give appropriate help and teach students how to analyze and solve problems when solving application problems.
Students should be given more opportunities to describe their thinking process in class. You can also organize students to speak in groups. By telling each other with your classmates, it is easy to cultivate students' ability to check and adjust their own thinking, so that their thinking and language expression skills can develop rapidly. With the increase of grades and the development of students' abstract thinking, students can think independently, evaluate each other, express different opinions, activate their thinking and pay attention to cultivating students' organized and grounded thinking. For example, in middle-grade teaching, x+5= 12. After students work out "x= 12-5, x=7", they can ask "What is the basis of your calculation?" Teaching 25× 13×4 requires students not only to talk about simple algorithms, but also to talk about basics. Also pay attention to the logical rigor of students' judgment. For example, when teaching divisors and multiples in senior grades, you can ask, "12 is divisible by 3, so we say that 12 is a multiple and 3 is a divisor. Is this judgment right? " Students should explain the reasons after answering. In short, we should attach importance to students' thinking process in teaching, but we should put forward different requirements according to students' age characteristics to gradually improve students' thinking ability.
Attach importance to the cultivation of thinking quality
The agility of thinking should be cultivated from the lower grades. For example, when teaching oral arithmetic, we should gradually put forward appropriate speed requirements. Teach students a calculation method, and after some practice, guide students to simplify the thinking process and further improve the calculation speed. For example, after teaching 9 plus a few and 8 plus a few, students can be guided to observe and compare, find out the changing law of numbers with the second addend, and then think about how to calculate numbers quickly. To cultivate the agility of thinking, we should pay attention to the appropriate requirements, give students time to think when asking questions, and don't make students too nervous.
In the process of applying knowledge to solve mathematical problems, teachers should pay attention to cultivating students' habit of "self-reflection". Because the development of students' self-awareness is still immature, they often ignore their internal psychological activities and are not easy to notice the defects and mistakes in their thinking. Therefore, in the process of organizing exercises, we should always guide students to reflect on their own thinking, consciously express their thinking process, and consciously test. In addition, the training of multiple-choice questions is also conducive to the critical development of thinking. Compared with other types of multiple-choice questions, the formulation of questions has changed. Although the topic is not big, it involves a wide range of contents and many traps. If you want to choose the right answer, you must think critically.
3 How to cultivate students' thinking ability
Encourage cooperation and exchange, and promote thinking.
Thinking and language are closely related. Einstein said: "The development of a person's intelligence and the way he forms concepts depend largely on language." Thinking is an indirect and general reflection of objective things. Although language is the shell of thinking, language itself has the functions of generality and indirectness. Without these functions, human thinking, especially abstract thinking, will be difficult. The ancients said, "If you have a will, you will have a word." "Speaking" is inseparable from the brain's thinking, and can promote the brain's thinking. In class, we often find that some children are always in a daze when describing the idea of solving problems, some children are unwilling to say it, some children say it incompletely, and so on, which often makes us feel very distressed. Therefore, in the process of mathematics classroom teaching, teachers should actively create a democratic and harmonious classroom atmosphere, let students dare to speak and be willing to speak, constantly provide students with opportunities to speak and encourage students to exchange ideas with their classmates.
For example, in the math practice class of "What's the circumference" in the third grade, a set of irregular figures are displayed in the "quantity and quantity" section, requiring students to measure and calculate the circumference. So I let the students think independently, measure the length of several sides before starting work, then exchange ideas in the group, and then discuss with each other before and after, so that the students who didn't think of using the translation method can be inspired, and then let the students report in class and get the following methods: just measure the length and width of the rectangle. In this way, the calculation of the perimeter of the original irregular figure is simplified, and students can appreciate the charm of mathematical thinking and master a good thinking method. Another example is the teaching of problem-solving strategies. An example: "Xiaoying Village used to have a rectangular fish pond with a width of 20 meters. Later, due to the expansion of the highway, the width of the fish pond was reduced by 5 meters, so the area of the fish pond was reduced by 150 square meters. What is the area of the fish pond now? " After the students found the conventional solution by drawing pictures, I asked, "Do you have a better solution besides this one?" Guide the students to rethink through the pictures they draw, and then exchange ideas with their deskmates. Subsequent teaching was brilliant, and different solutions emerged one by one:150 ÷ 5× 20-150; 20÷5× 150- 150; (20÷5- 1)× 150。 Starting from the quantitative relationship and the characteristics of numbers, students have obtained many new solutions. I successfully played a listener here, giving students enough time to think and communicate, giving full play to their subjective initiative and taking care of their findings one by one. Almost every solution is born, and every step of teaching is deepened, which hides words full of encouragement and trust: "Do you have a better solution? Share your thoughts with your classmates! " "Your idea is really unique!" A practical problem solved by drawing produces a novel thinking spark under the action of students' individual initiative, which avoids the mechanization and simplification of thinking. Students have realized that "learning knowledge" and "speaking knowledge" are happier and more successful than "listening to knowledge".
Carefully design questions and guide students' thinking
Cultivating students' thinking ability is the same as learning calculation methods and mastering problem-solving methods, and thinking is closely related to the problem-solving process. The most effective way to cultivate thinking ability is through problem-solving practice. Therefore, designing exercises well has become an important part of improving students' thinking ability. Teaching should be adjusted or supplemented according to the specific situation.
Pupils have poor independence and are not good at organizing their own thinking activities. They often think about what they see. Cultivating students' logical thinking ability is mainly through the demonstration, guidance and guidance of teachers in the teaching process, so that students can acquire some thinking methods in a subtle way. Teachers carefully design questions in the teaching process, put forward some enlightening questions, stimulate thinking, and mobilize students' enthusiasm and initiative to the maximum extent. Students' thinking ability can be effectively developed only when they are active in thinking. First of all, design exercises should be targeted and designed according to the training objectives. Secondly, design a variety of practice forms. Through various forms of practice, it is not only helpful to deepen the understanding of the mathematics knowledge learned, but also helpful to develop the flexibility of students' thinking and stimulate their interest in thinking. In short, in the teaching process, teachers should put forward exercises with moderate depth according to the key points of the textbook and the actual situation of students.
How to Cultivate Students' Mathematical Thinking Ability
Cultivate application consciousness and deepen thinking
Everyone learns useful mathematics, everyone uses useful mathematics, and it is our teaching goal to cultivate students' ability to solve practical problems by using mathematical knowledge. Students should not only master knowledge, but also learn to apply it. Only in this way, mathematics will be smart, full of vitality, and the value of mathematics can be truly realized. When students can think about the problems they encounter from the perspective of mathematics and find strategies to solve them, they will definitely re-create and process the knowledge they have learned and promote the in-depth development of thinking. Therefore, it is particularly important to cultivate students' application consciousness from an early age. For example, in the fourth grade, there is a practical activity in the textbook. Before class, I divide the students into good groups and arrange the materials brought by each group. In class, I first demonstrate experiments in the classroom, clarify the norms and essentials of experimental operation, and then lead students to the playground for activities. As a result, only the data of two groups of students were unified, and the answers of other groups were different. Many students have raised their own doubts: Teacher, why can't we get a unified result in our experiment? Is such an experiment meaningful? Why are there so many different results? What other factors affect the rolling of this object? This series of questions shows that the application of mathematical knowledge can make students' thinking develop in depth, and can constantly inspire students' thinking and deepen it.
For another example, after students have learned simple statistical knowledge and mastered the method of recording data by orthography, in order to let students experience the whole process of statistics and appreciate the application value of statistics, I arranged an extracurricular survey: the class book corner is going to buy some new books, which books will be welcomed by everyone? When solving this practical problem, students can take the initiative to use what they have learned, look for strategies to solve the problem from the perspective of mathematics, and truly feel the great value of mathematics application in life in the activities.
Pay attention to strengthening the thinking of solving problems.
In teaching, our teachers should guide and cultivate students to form the habit of analyzing and solving problems in the whole process. At the beginning of solving a problem, students should be guided to effectively estimate and judge the structure, nature and difficulty of the problem, as well as the relationship between the problem and the previous problem solving, so as to ensure that the problem solving follows the correct, meaningful and even the best thinking route; In solving problems, students should be guided to adjust their thinking process and direction at any time according to the progress and requirements of solving problems; After solving the problem, students should be guided to check whether the expected purpose has been achieved and consider whether there is a better solution.
The conclusion of traditional application problems is that students are often satisfied with finding an answer without further thinking and analysis. Designing application questions with open conclusions can cultivate students' innovative spirit. For example, three engineering teams, A, B and C, jointly built a canal with a contract capital of 6,543,800 yuan. After the joint repair of the three teams was completed 1/3, team A left, two-thirds of team B stopped working, and team C completed the final 1/3 alone. How much do the three teams get each? I gave students enough time to think and practice, explore a more reasonable distribution method, and let students solve practical problems independently. Through discussion, the students have the following ways to solve the problem: (1) Starting from 1/3, divide 600,000 yuan into three teams, each with 200,000 yuan. In the middle 1/3, teams B and C each get 300,000. Finally, 1/3 C completed it alone and got 600,000 yuan. (2) According to the canal length ratio 1: 2: 3 completed by Team A, Team B and Team C, the share of Team A is180x1(1+2+3) = 300,000 yuan, and that of Team B is/kloc-. (3) Take the average of the results of (1) and (2). In this way, students use different strategies to solve the same practical problem and get different results, which effectively promotes students' independent inquiry.