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How to Cultivate Students' Mathematical Analysis and Problem Solving Ability
The problem is the wave of classroom teaching, which is the prelude to the climax of classroom teaching. Only when there are "problems" can there be unknowns and the development of teaching activities. The great scientist Einstein once said: "The generation of a problem is usually more important than the conclusion. Because the conclusion just needs some mathematical or experimental skills. It takes creative imagination to put forward new problems, new possibilities, or analyze an old problem from a new angle. Only in this way can science really progress. " Therefore, it is particularly important to guide students to ask questions scientifically and cultivate their problem consciousness.

How to make students ask questions of interest and scientific research value in science class, and make students' questions really become the starting point of inquiry? I think teachers should first establish a sense of problems, dare to let students ask questions, dare to let students "ask questions", be good at raising students' questions and have exchanges and discussions, which is also strongly advocated by science teaching. Of course, the cultivation of problem consciousness cannot be achieved overnight, and it needs a gradual process. First, students want to ask questions, then students want to ask questions, and finally students should be good at asking questions. Teachers should consciously cultivate students' problem consciousness in the process of "thinking, knowing and being good at asking questions", and indirectly cultivate and create students' innovative thinking and scientific exploration spirit.

First, guide students to "want to ask" and sprout the desire to question and explore.

In science class, although students are curious about the world and have a desire to explore knowledge, a considerable number of students do not want to ask questions, do not like to ask questions, dare not ask questions, and are indifferent to "questions", which requires the encouragement and guidance of teachers. The typical guidance method is to create a situation and create suspense, so that students will be skeptical until they have to ask, and then ask questions quickly. For example, when guiding students to discover the capillary phenomenon of water, the teacher prepares a glass of red water, then cuts a thin paper towel and puts the lower part of it into the water. After a while, he asked, "What do you see?" The students happily replied, "The water will climb up along the paper towel" and "The paper towel is dyed red". Some students can't wait to blurt out: "Why does water climb up along the paper towel?" "How did the water get up?" "If you change a note or something, will the water climb up?" ..... because of the suspense of "water climbs up along the paper towel", a series of questions have been raised by students. For another example, in the teaching of "Putting Solids into Water" in the second volume of the third grade, when studying the properties of solids, teachers show common objects in life, such as pencils, erasers, leaves, spoons, marbles, foam plastics, iron blocks, whole apples and whole candles, so that students can predict which of these solids will sink and which will float when put into water. After the students predicted, the teacher asked, "What did you predict?" The student replied, "The heavy ones will sink, while the light ones will float.". Then the teacher demonstrated that a small stone sank in the water and a big ship floated on the water. Because of the creation of the previous situation and the image demonstration of the following courseware, students' "original cognition (heavy will sink, light will float) will conflict with reality", and suspense will arise, and the problem will follow: "Light pebbles will sink, but heavy boats will float on the water. Why? "

Second, guide students to "ask questions" and learn how to ask questions and ask difficult questions.

When students want to ask, it means opening the floodgate of thinking, which is the beginning of asking questions. But just because you want to ask and dare to ask doesn't mean you can ask. Many students don't know how to ask questions. The questions they asked were childish, superficial and worthless. Sometimes they will ask questions that have nothing to do with the learning content, waste teaching time, even affect the thinking direction and interfere with the exploration and solution of problems. This is not "knowing perfectly well past asking". For example, the above example guides students to "discover the capillary phenomenon of water", and some students ask "What's in the red water?" "Why is the water red instead of green?" , and so on. Obviously, "knowing how to ask questions" should be of practical and exploratory value, which involves the method of asking questions and the method of asking difficult questions. So how to guide students to "ask questions"? First of all, teachers should carefully design questions, demonstrate and guide students to ask questions, so that students know where to ask, what to ask and how to ask. Secondly, we should teach students the skills and methods of asking questions. For example: ask questions around the topic; Ask questions around key and difficult points; Ask questions from research methods; Ask questions according to your own guess; Ask questions at any time according to the experimental process; Ask questions according to the phenomena they observed; Ask questions according to possible results, etc. If the questions raised by students are still not targeted, or deviate from the questions explored in class, what teachers should do to solve this problem is to classify the various questions raised by students, which can be merged, which belong to in-class research, which belong to post-study and which can be solved on the spot, and teachers must explain them clearly to students. For example, in the activity of leading students to blow bubbles, the teacher demonstrated and asked: Why are bubbles colorful in the sun? How can I blow out a big bubble that is not easy to break? If you don't use a straw (change the shape of the tool for blowing bubbles), will the bubbles still be spherical? What's in the water blowing bubbles? ……

Only in this way can students master the basic thinking methods of finding and asking questions, see anomalies from the ordinary and find special from the ordinary, thus constantly exploring knowledge and cultivating innovative spirit.

Third, guide students to be "good at asking questions" and sublimate the consciousness of questioning and solving doubts.

When students "know how to ask questions", they basically have a certain sense of problems and have a certain ability to find and solve problems. However, science teaching should not be limited to this, but should also actively guide students to change from "being able to ask questions" to "being good at asking questions". The so-called "good questions" should not only learn some methods of asking questions and asking difficult questions, but also be able to ask questions from multiple angles, keenly grasp the essence of the problem, dare to put forward different opinions and ask valuable questions. "Being good at asking questions" is the highest realm of cultivating problem consciousness, the pursuit of students, and what teachers should strive to create. How to guide students to be good at asking questions, and finally achieve the sublimation of asking questions and asking difficult questions? We can do the following: 1. Guide students to grasp contradictions and ask questions. For example, the lens of eyeball is equivalent to a convex lens, and the image made by convex lens is upside down. Why is the scene we see with our eyes positive? The second is to guide students to use hypothetical conditions to ask questions, such as: If the road to the top of Baiyan Mountain has only a slope instead of a spiral, is it easy for people to climb it? Why? The third is to guide students to learn to ask questions in reverse thinking, such as: What will happen if the rotation direction of the earth changes from east to west? If the tower is built with a big top and a small bottom, is it stable? Why?

These three methods of asking questions and asking difficult questions should be said to be "good at asking questions", which can well promote students' innovative thinking and the development of innovative thinking. After students "know how to ask questions", teachers must pay attention to making students "good at asking questions" on this basis.

In a word, it is very important to cultivate students' problem consciousness in science teaching. To cultivate problem consciousness, we should start with guiding students to "want to ask", "know how to ask" and "be good at asking", and combine life practice and scientific social practice to "ask" innovation, "ask" value and "ask" excitement.

One of the general goals of the new curriculum standard is to make students "have innovative consciousness, be able to think independently, be brave in doubt, and cultivate a scientific attitude and spirit of respecting facts and boldly imagining." "Innovative consciousness, scientific attitude and scientific spirit" are very important for students no matter what they do in the future. The establishment and development of physics embodies people's scientific attitude, scientific spirit and innovative consciousness everywhere. Therefore, physics course has unique advantages in cultivating students' quality. More opportunities should be created in physics teaching activities to promote the realization of this goal.

In the process of using new physics textbooks, how to find a breakthrough in innovative education, give full play to students' main role and cultivate students' innovative spirit and practical ability according to the characteristics of physics subjects? I think we should start from the following aspects:

First, stimulate students' interest and curiosity in learning

Innovative activities need innovative motivation to stimulate and maintain. Because intrinsic motivation is formed when individual evaluation matches their own interests, intrinsic motivation is directly related to interests in the process of innovation. Interest is a great motive force for a person to acquire knowledge and develop ability, which can make students become proactive, full of confidence, active in exploration and brave in innovation. For example, I demonstrated the experiment of supporting table tennis with a hair dryer in the teaching of the relationship between flow rate and pressure. Turn on the hair dryer and blow it up. Put a ping-pong ball at the air outlet of the hair dryer so that it won't fall off. While moving the hair dryer left and right, the students happened to see table tennis moving with the hair dryer, either fast or slow. This phenomenon greatly surprised the students, making the classroom atmosphere active at once, and they immediately developed a strong thirst for knowledge without the need for teachers to talk more. The forerunner of interest is curiosity, which is the potential and sprout of innovative consciousness. People with innovative spirit are often very curious. Einstein once said that he had no special talent, only a strong curiosity. It is worth noting that curiosity will gradually decline and wither if it is not cultivated and supported. Therefore, when students are driven by curiosity to ask various questions or do something irregular, as teachers, they should be encouraged on the basis of careful investigation. In the teaching of "preliminary circuit exploration", when it comes to connecting the two poles of power supply, it is absolutely not allowed to connect directly with wires to avoid short circuit and damage to power supply. Naughty students deliberately short-circuited two batteries. At this time, they should not be reprimanded and denied. Instead, let them touch the copper wire of the conductor and guide them to feel the feeling of their hands. They will find their hands very hot. Then tell them that this is the thermal effect of current. Students not only consolidated their knowledge, but also learned new knowledge and satisfied their curiosity. In this way, they prefer to do things and ask questions in class, and their sense of innovation has been strengthened.

Second, attach importance to physical experiments and cultivate innovative consciousness in scientific inquiry.

Physical experiment is a scientific method for people to know and study natural phenomena. It is a process that people creatively use scientific knowledge and experimental means to carry out research and practice activities in a planned way according to the research object and purpose. Therefore, the experiment process itself is a creative inquiry process. The history of the development of physics fully proves that any major breakthrough in physics is achieved through experiments without exception. As Boyle said, it is impossible to know anything new without experiments. When experiment is used as a way of learning, the creativity and inquiry of experiment are fully demonstrated. To achieve the goal of physics classroom experiment, you don't have to have great inventions and discoveries like scientists. The discovery of new problems, new knowledge, the formation of new ideas and new methods in physical experiments are similar to the research process of scientists on science. At the same time, through the experience and understanding of the experimental process, it is a kind of gain to eliminate the experimental faults and let students realize the hardships and success of creation. Moreover, the purpose of the experiment is not only the result. It is good to draw valuable conclusions, but it is more important for students to broaden their horizons, expand their thinking, pay attention to phenomena, develop their personality and cultivate their abilities during the experiment. Students can experience the process and method of scientific research through their own initiative, and get the pleasure of scientific inquiry and the joy of success. Therefore, in the teaching of new physics courses, making full use of experimental means to learn new courses is conducive to cultivating innovative consciousness.

Third, create a relaxed environment to induce students to expand their imagination.

Students who study physics for the first time are generally interested in physics, because physical knowledge and physical phenomena come from life, which can satisfy students' curiosity, and many physical experiments can make students think. But before long, some students will become numb and even disgusted with the physical phenomena and knowledge that teachers think are wonderful. This is because the teacher didn't satisfy the students' curious questions in time when explaining this knowledge, which broke the wings of students' association and put students' thinking into the knowledge framework that they didn't want to enter.

When Einstein was asked why he could create, he said, "I have no other talent, just like asking questions." Therefore, to cultivate creative thinking ability, teachers should first change their teaching concepts, give way to students' learning in teaching, and be partners and guides of students' learning, not masters. Tolerate others, let students participate in teaching equally, let students overcome psychological obstacles, boldly question and ask difficult questions. Secondly, we should protect students' curiosity and create sparks in teaching, and don't look at students' world with adult behavior and mature eyes. Use heuristic teaching to guide students to seek differences and questions, encourage students to ask more questions and encourage them to think independently as researchers and creators.

Fourth, enrich extracurricular activities and let students experience the joy of success.

Extracurricular scientific and technological activities are an effective position to enrich students' spiritual life, broaden their horizons, cultivate their sentiments and stimulate innovation. It provides good intellectual nutrition and good mood and environment for the formation of creative thinking ability. In this activity, students can develop independently without being bound by the scope of teaching materials and teachers' inclination. Therefore, extracurricular activities should be rich in content, novel, diverse in forms and flexible in methods. For example, holding a science and technology production competition to mobilize students to make small inventions, small productions and small creations. Arrange appropriate extracurricular exploration experiments every week, and conduct small experiments with things around you, so that students can think more, do more, think more and do more, and be inspired by them, so that students can observe the truth that they have never seen in classroom teaching in knowledge, science and interest activities, understand the mystery that classroom teaching is too late, and touch what they can't get in classroom teaching. Through small-scale production, small experiments and other physical practice activities, students are provided with more opportunities for hands-on, brains and self-development, which can fully tap students' potential innovative literacy, continuously improve students' ability to apply knowledge, and develop students' personality, specialty and innovative skills.

In a word, physics teaching is not only to impart knowledge, but also to guide students to think, inspire students' creative thinking and cultivate students' creative ability. In teaching practice, as long as we strive to explore and innovate, organically combine learning with creation, imitation with innovation, and rationality with fantasy, we will be able to effectively develop students' creative potential and cultivate students' creative thinking ability. It has been a long time since the implementation of the new curriculum standards, which has brought many changes, disputes and explorations, and also promoted the continuous development of education. Under the requirements of the new curriculum standards, the purpose of middle school mathematics teaching is to cultivate students' problem-solving ability in the final analysis, and improving mathematics problem-solving ability is a very important task in mathematics teaching. Improving students' problem-solving ability runs through teaching and must be placed in a very important position. Then, how to improve students' problem-solving ability, the specific methods can mainly start from the following aspects:

First, cultivate the ability of combining numbers with shapes.

In the learning process of high school mathematics, numbers and shapes are everywhere. There are two branches of junior high school mathematics-algebra and geometry. Algebra studies "number" and geometry studies "shape". But algebra should be learned by means of "shape" and geometry by means of "number". "The combination of numbers and shapes" is a trend. The more you learn, the more inseparable you are from "number" and "shape". Especially after the establishment of plane rectangular coordinate system, the study of function can not be separated from images. Often with the help of images, the problem can be clearly explained, and it is easier to find the key to the problem, thus solving the problem. This is not only intuitive, but also comprehensive, easy to find the breakthrough point, which is of great benefit to solving problems.

Second, enhancing self-confidence is the key to solving problems.

Self-confidence can make you stand on your own feet. In the exam, I always see that some students have a lot of blanks in their papers, and many questions have not been done at all. As the saying goes, artists are bold. You don't have much courage without high art, but not doing it is one thing, and not doing it is another. A slightly more difficult math problem, it is impossible to see the answer and result at a glance. It is necessary to analyze, explore, draw, write and calculate. Only through tortuous reasoning or calculation can we reveal some connection between conditions and conclusions, and the whole idea will be clear. How do you know you won't do it if you don't do it? Even a teacher can't answer you immediately when he meets a difficult problem. It is necessary to analyze and study, and find the correct way of thinking before teaching. Dare not do a little more complicated questions (not necessarily difficult questions, some questions are just a little more narrative), which is a sign of lack of confidence. Self-confidence is very important in solving mathematical problems. Believe in yourself, as long as you don't go beyond your knowledge, you can always solve any problem with what you have learned. Dare to do problems and be good at doing them. This is called "strategic contempt for the enemy, tactical attention to the enemy." When solving a specific problem, we must carefully examine the problem, firmly grasp all the conditions of the problem, and don't ignore any one. There is a certain relationship between a problem and a class of problems. We can think about the general idea and general solution of this kind of problem, but it is more important to grasp the particularity of this problem. Grasping the difference between this problem and this kind of problem, there are almost no similarities in math problems, and there are always one or several conditions that are different, so the thinking and problem-solving process are different. Some students and teachers can do the problems they talk about, but others can't. They just talk about things, and there is no way to start with some small changes in the topic. Of course, where to start first is a tricky thing, and you may not be able to find it accurately. But it is absolutely right to grasp its particularity when doing the problem. Choose one or several conditions as the starting point to solve the problem, and see what can be drawn from this condition. The more you get, the better, and then choose other conditions for calculation or calculation. There are many solutions to common problems. I believe that using the conditions of this problem, combined with the knowledge I have learned, will definitely draw a correct conclusion. The topics of mathematics are infinite, but the ideas and methods of mathematics are limited. As long as we learn the basic knowledge well and master the necessary mathematical ideas and methods, we can successfully deal with endless problems. The topic is not to do more, the better. The ocean of topics is endless, and you will never finish reading it. The key lies in whether you have cultivated good mathematical thinking habits and mastered the correct mathematical problem-solving methods. Of course, doing more questions has several advantages: First, practice makes perfect, speeding up the pace and saving time, which is especially important in the case of limited examination time; The second is to consolidate and memorize the definitions, theorems, rules and formulas learned by doing the problems, thus forming a virtuous circle. Solving problems requires rich knowledge and more confidence. Without self-confidence, you will be afraid of difficulties and give up. Only by self-confidence can we go forward, not give up easily, and study harder, can we hope to overcome difficulties and usher in victory.

Third, cultivate the thinking ability of "correspondence"

The concept of "correspondence" has a long history. For example, we correspond a pencil, a book and a house to an abstract number "1", and two eyes, a pair of earrings and a pair of twins to an abstract number "2". With the deepening of learning, we extend correspondence to a relationship, a form and so on. For example, in calculation or simplification, we will correspond to the left x of the formula, corresponding to a; Y corresponds to b; Then use the right side of the formula to get the original result directly. This is to use the idea and method of "correspondence" to solve problems. Grade two or three will see one-to-one correspondence between points on the number axis and real numbers, one-to-one correspondence between points on the rectangular coordinate plane and a pair of ordered real numbers, and the correspondence between functions and their images. The thought of "correspondence" will play an increasingly important role in future research.

Fourthly, cultivate students' mathematical "transformation" thinking ability.

The most fundamental way to solve mathematical problems is to "turn the difficult into the easy, simplify the complex, and turn the unknown into the known", that is, through certain mathematical thinking, methods and means, a complex mathematical problem is gradually transformed into a well-known simple mathematical form, and then it is solved through familiar mathematical operations. For example, if our school wants to expand the campus area, it needs to requisition land from the town. The town gave an irregular piece of land. How to measure its area? Firstly, the actual terrain is drawn into paper graphics with a small flat instrument (level or theodolite if possible), then the paper graphics are divided into several trapezoid, rectangle and triangle, and the sum of the areas of these graphics is calculated by the learned area calculation method, and then the total area of this irregular terrain is obtained. Here, we transform the irregular graphics that can't be calculated into regular graphics that can be calculated, thus solving the problem of land survey. In addition, all kinds of multivariate equations and higher-order equations mentioned above can be finally transformed into linear equations or quadratic equations with one variable by means of elimination and simplification, and then solved by known steps or formulas. Transforming thinking is the most important thinking habit in solving problems. In the face of difficult problems and unfamiliar problems, we must first think of transformation, which can always be transformed. Usually, teachers should pay more attention to how to solve problems, and how to "make the difficult easy, simplify the complicated, and turn the unknown into the known". Students should also exchange more experiences of successful transformation, deeply understand the true meaning of transformation, and effectively master the thinking and skills of transformation.

Fifth, cultivate the thinking ability of "equation"

Mathematics studies the spatial form and quantitative relationship of things. The most important quantitative relationship is equality, followed by inequality. The most common equivalence relation is "equation". An equation containing an unknown quantity is an "equation", and the process of finding the unknown quantity through the known quantity in the equation is to solve the equation. We were exposed to simple equations in primary school, but in the first year of junior high school, we systematically studied the solution of one-dimensional linear equations and summarized five steps of solving one-dimensional linear equations. If you learn and master these five steps, any one-dimensional linear equation can be solved smoothly. In the second and third grades, they will also learn to solve quadratic equations in one variable, quadratic equations in two variables and fractional equations. In high school, we will also learn exponential equation, logarithmic equation, linear equation, parametric equation and polar coordinate equation. The solution ideas of these equations are almost the same, and they are all transformed into linear equations or quadratic equations with one variable by a certain method, and then they are solved by the familiar five steps to solve linear equations with one variable or the root formula to solve quadratic equations with one variable. Energy conservation in physics, chemical equilibrium formula in chemistry, and a large number of practical applications in reality all need to establish equations and get results by solving them. Therefore, students must learn how to solve one-dimensional linear equations and two-dimensional linear equations, and then learn other forms of equations well. The so-called "agenda" thinking means that for mathematical problems, especially the complex relationship between unknown quantities and known quantities encountered in reality, we are good at constructing relevant equations from the viewpoint of "equations" and then solving them.