1 how to cultivate students' innovative thinking in mathematics
Dare to let go, dare to let students explore boldly and cultivate students' open thinking.
Open teaching has become a hot spot in basic mathematics education, item teaching in senior high school entrance examination and mathematics teaching reform and research. Open-ended questions are characterized by incompleteness, uncertainty, divergence, exploration, development and innovation, and their answers are also unfixed, unnecessary, uncertain and unsolvable. To cultivate students' open thinking in classroom teaching is to select examples, give priority to inspiration, be concise, guide and prompt more, give students enough time to think about problems, let students explore boldly, fully mobilize their thinking enthusiasm and improve their thinking quality.
For example, there is a problem in algebra in grade three, which passes through the point (1, 2) and the resolution function of y increases with the increase of x, right? Shake it? Shake it? Shake it? Shake it? Shake it? Shake (just write one). The result of this question is negative, but there are only two conditions: ① uniform y increases with the increase of x; ② Passing point (1, 2). Only linear functions and proportional functions that meet the condition ① can be set up with their analytical expressions, and then students can get the forms of y=2x, y=x+ 1 and y=4x-2 through exploration.
Strengthen thinking training, analyze the essence of the whole problem and improve students' overall thinking.
Cultivate students' overall thinking ability, that is, cultivate students' mathematical induction and summary ability, so that they can form good thinking habits in the learning process, be willing to deal with problems, and truly master knowledge. Therefore, in teaching, teachers must pay attention to more guidance and give students more opportunities to summarize and summarize independently.
For example, in the review of the chapter on rational numbers, you can ask: "How many concepts and rules are there in the result of 0?" Students will get the following answers after thinking: ① The inverse of 0 is 0; ② The absolute value of 0 is 0; ③ The sum of two opposite numbers is 0; (4) Any number multiplied by 0 will get 0; ⑤ Divide zero by any number that is not zero to get 0; ⑥ Multiply several rational numbers by 0 to get 0. In this way, students will have an overall understanding of 0.
2 the cultivation of mathematical thinking
Strengthen reflection and improve students' application ability
On the one hand, reflection and summary in learning can help students better review their learning process, on the other hand, students can find their own areas that need improvement in reflection. Reflecting on the learning content in the preview stage can enable students to preview more effectively in the future, and also enable students to better understand related issues. Reflection in teaching analysis stage is of great help to improve students' mathematical thinking and logical ability. Reflection in the training stage allows students to review what they have learned in the process of reviewing the answers to a certain type of questions, so that students can find out the answering skills and specific methods of a certain type of questions in the long-term thinking. Therefore, these have an important influence on the cultivation of students' ability and the development of mathematical thinking.
For example, in the teaching of analysis, the relevant content of quadratic function is used to solve examples. In the process of reflection, students will first analyze the relevant conditions involved. "The purchase price of each piece is 8 yuan, and the price is 10 yuan. You can probably sell 1 10 pieces a day. If the unit price of goods is reduced by 0. 1 yuan, the sales volume can be increased by 65438. What do these conditions have to do with the required maximum profit? In the "five steps" of the analysis stage, the relationship between each step is gradual, which is a very meticulous logical thinking. Finally, on the basis of such a step related to reflection, it seems that students are reviewing this topic, which is actually a summary of the specific application of quadratic function. Once students discover this rule, they will find practical problems related to quadratic function. The general steps to solve the problem are: defining the known conditions-determining what the problem needs to solve, whether to seek the maximum value or something else-how the known conditions relate to the problem-what is the potential established range-listing the analytical formulas to solve according to all the excavated conditions.
Guiding analysis and cultivating students' comprehensive ability
In mathematics teaching, students' dominant position should be fully highlighted, that is to say, mathematics teaching should deepen knowledge and combine with reality. Teachers should pay attention to the logical level of teaching methods and cultivate students' logical ability and thinking in example teaching.
The process of teacher's analysis is actually the process of guiding students to logically analyze and sort out problems. In this process, students will continue to improve, and the level and ability of logical thinking will continue to strengthen. In the long run, students' logical thinking ability can be improved to some extent. Of course, in this process, especially in the analysis link, teachers can also take the way of guiding questions and answers to mobilize students' participation, highlight students' dominant position and enliven the classroom atmosphere.
3 the cultivation of mathematical thinking
Attach importance to practical operation and cultivate the subject's exploration ability
Operation practice is an important way to cultivate students' creative thinking. Let students take the initiative to operate, let students break through the barriers of time and space, acquire perceptual knowledge that is lacking in life and must be mastered, turn abstraction into image and knowledge into ability. Let students get a real understanding happily in their own creation, and effectively cultivate their exploration ability.
For example, when I was teaching "How many symmetry axes does a circle have", I asked each student to cut out a circular piece of paper and fold it freely. In the process of folding, I asked the students to observe carefully and operate constantly, and naturally came to the correct conclusion. Not only the thinking process is fully exposed, but also the students take the initiative to learn and truly internalize social knowledge into their own individual knowledge. Another example is to guide students to operate when teaching "28+7" oral arithmetic. "What if 8 sticks +7 sticks exceed 10?" Pay attention to guide students to find solutions to problems, and then ask students to summarize and derive calculation methods according to the appearance of the joystick, and actively understand the calculation, which is intuitive and effective. Also pay attention to let students dictate their own operation process, guide students to sum up arithmetic, and make operation, thinking and expression constitute a complementary internalization process. Through practical operation activities, the algorithm of "from ten digits to ten digits into one" is explored.
Stimulate subjective initiative and cultivate innovative subject
As a teacher, in teaching, we must always think about mathematical procedures from the perspective of students, consider the classroom structure, and regard students as real learning masters. Let students learn vividly, actively and effectively, and let all students actively participate in the whole process of learning from beginning to end. Children who have just entered school are curious, active and competitive. They are eager for knowledge, willing to participate in various activities, and like to study new problems and discover new laws.
I grasped the psychological characteristics of students in the teaching of "combination of oral calculation and written calculation", and introduced learning tools into the classroom as soon as I entered school, which made students full of curiosity and freshness. I'll teach them how to operate first. For them, sticks and numbers are not only learning tools and calculating tools, but also "toys". When they know that these learning tools can help them learn math well, they are deeply attracted. The free pendulum, collective pendulum and group competition pendulum in the classroom have both the color and atmosphere of the game. Students put sticks and numbers, act quickly and have high interest. Sometimes looking at sticks, sometimes listening to numbers, and sometimes stating the operation process, the coordinated activities of eyes, ears, hands, mouth and brain conform to the psychological characteristics of children's single attention, forming a wide range of information channels, which makes their thinking extremely excited. At the same time, oral calculation, written calculation and estimation are combined, and learning, practice and application are carried out alternately, so that the excitement and inhibition of children's brain nerves can be adjusted each other, the learning mood is high, the atmosphere is active, and learning is entertaining, which meets children's psychological requirements to a certain extent, arouses their strong interest and mobilizes their enthusiasm and initiative in learning mathematics.
4 the cultivation of mathematical thinking
Methods of teaching students to think
Modern education view holds that mathematics teaching is the teaching of mathematical activities, that is, the teaching of thinking activities. How to cultivate students' thinking ability and develop good thinking quality in mathematics teaching is an important subject of teaching reform. Confucius said: "Learning without thinking is useless, thinking without learning is dangerous". In order to make students think actively in mathematics learning, we must teach them the basic methods of analyzing problems, which is conducive to cultivating students' correct thinking mode. To be good at thinking, students must attach importance to the study of basic knowledge and skills. Without a solid foundation, their thinking ability cannot be improved.
Mathematical concepts and theorems are the basis of reasoning and operation. In the teaching process, we should improve students' cognitive ability of observation and analysis, from outside to inside, from here to there; In the example class, the discovery process of solving (proving) problems should be regarded as an important teaching link, so that students should not only know how to do it, but also know why and what prompted you to do it. In mathematics practice, we should carefully examine the questions, observe them carefully, have the ability to dig out the hidden conditions that play a key role in solving problems, and use comprehensive methods and analytical methods to express them in mathematical language and symbols as much as possible in the process of solving problems (proofs).
There is no standard answer, and differences are encouraged.
1. Encourage students to find different ways to accomplish the same task. For example, when solving the problems recorded on the tombstone of Diophantine, the Greek mathematician, first let the students discuss in groups how to list the equations. When the students list the equations, see who can give the answer the fastest! A classmate gave the correct answer: 84. He said: I think people's age should be a positive integer, and this positive integer can definitely be divisible by every denominator in the equation. The least common multiple of the denominator of the equation is 84. So I think it's 84. Such exercises can stimulate students' thinking, thus improving their thinking ability.
2. Guide students to understand or express the same question in different ways. For example, when teaching the practical significance of algebra, students are encouraged to list as many examples as possible related to their own lives or their surroundings, but not less than three, not the same example. In this way, every student can have something to say and understand the practical significance of algebra better.