1 how to guide students' mathematical thinking
Use one question to solve many problems skillfully, think in many directions, and break through the mindset.
Teaching practice shows that to overcome the negative thinking pattern, we should start with changing the normal state of students' problem-solving thinking, break down the barriers between different problem-solving methods, find the connections between them, and inspire students to pay attention to these connections in use. Paying attention to multiple solutions to a problem is a good form to cultivate divergent thinking, which is conducive to the establishment of knowledge and a leap in understanding, and at the same time can expand the freedom of students' independent learning and create favorable conditions for improving their problem-solving ability. Flexible thinking mode is closely related to creative thinking. If a student only thinks and deals with problems in a fixed way or in a way taught by a teacher, he can't be creative.
Teachers should let students develop a habit and method of thinking from multiple angles, instead of sticking to one angle and one model, so as not to cause students' thinking methods to be single and rigid. In the usual teaching, we should encourage students to think from many angles and aspects, and constantly inspire students to think differently. Let students have a "discerning eye" in their thinking of seeking differences and see everything through the "fog" in order to explore more ingenious ways to solve problems. For example, when teaching the following examples 1 and 2, students can be guided to choose the best solution from the process of exploring different solutions.
Clever use of one topic is changeable, multiple topics are unified, and the mindset is broken.
"Mathematics is an ocean of problems", and teachers can't ask students to do all the math problems. Training students on various topics is an important means to consolidate basic knowledge and cultivate their ability, and it is also very important to cultivate the profoundness and broadness of students' thinking. In normal teaching, teachers can guide students to change the examples and exercises in textbooks in various ways.
Such as changing conditions, changing conclusions, changing data or graphs, extending conditions or conclusions, opening conditions or conclusions, or opening conditions and conclusions at the same time. By changing a problem and unifying the training of multiple problems, we can closely link the knowledge learned in each stage with all aspects of knowledge, deepen our understanding of knowledge and realize that mathematics is a whole, but more importantly, we can achieve the goal of solving a problem and understanding a class of problems, which can stimulate students' interest in learning, innovative consciousness and exploration spirit, cultivate students' innovative ability and learn to learn.
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
"Doing" Mathematics and Guiding Students to "Play"
Children's nature is to "play", and the new mathematics curriculum should also change the old-fashioned face and let students "play" well! "Playing" mathematics means that students perceive things through idiosyncrasies or materialized activities under positive emotional experience. With the game of problem consciousness, "play" has a direction. If "asking" is the starting point and main line of learning, then "playing" is the way to explore the main line. "Playing" mathematics is not only a cognitive process of students, but also a hunger process of teacher-student activities and emotional communication. Emotional activities belong to the dynamic system, which can promote the active participation of the subject. The uniqueness of "playing" mathematics lies in that the learning subject is in a happy and positive psychological state and actively and consciously "does".
Compared with passive "memorizing" mathematics, it changes "asking me to learn" into "I want to learn". "I want to learn" is based on students' internal learning needs, while "I want to learn" is based on external motivation and compulsion. The inherent needs of students in learning mathematics are mainly manifested in their interest in learning. There are direct interests and indirect interests. Direct interest points directly to the activity itself, and indirect interest points to the result of the activity. With students' interest in learning, learning activities are no longer a burden, but a kind of enjoyment and a quick experience, which makes students want to learn more, more willing to learn and more fond of learning. "Play" must be based on independent exploration and group cooperation. The "game" in this situation can make the classroom of mathematics learning a place for mathematics research and cooperation and communication between people, and can improve students' adaptability, cooperation and communication qualities necessary for adapting to the future society.
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.
4 the cultivation of mathematical thinking
Doing "Mathematics" and Guiding Students to "Use"
Mathematics will show its value and charm only when it returns to life. Only when students return to life and use mathematics can "everyone learn valuable mathematics" be truly realized. In mathematics teaching, teachers should be good at collecting teaching examples in real life and introducing themes from social life into mathematics classroom teaching, so that students can establish the consciousness of "using mathematics", cultivate their ability and experience the fun of "using mathematics" in the process of discovering problems, solving problems and practicing activities. It is also necessary to guide students to find mathematical problems from real life and establish the consciousness of "using mathematics".
Mathematics comes from life, and there is mathematics everywhere in life. "Mathematics Curriculum Standards" points out: "In teaching, we should create learning scenes that are closely related to students' living environment and knowledge background and are of interest to students, so that students can gradually understand the process of the generation, formation and development of mathematical knowledge in activities such as observation, operation, speculation, exchange and reflection, gain positive emotional experience, feel the power of mathematics and master the necessary basic knowledge and skills. "Let students participate in some practical activities with math problems, and experience the fun of' using math' while improving their ability. In mathematics teaching, teachers organize students to participate in mathematics practice activities with real life background purposefully and in a planned way, and solve some simple practical problems by using the mathematics knowledge they have learned, which can not only consolidate the mathematics knowledge they have learned, but also broaden students' mathematical horizons.
Doing mathematics to guide students to "realize"
Enlightenment is an important stage of mathematics and any other learning.
"Enlightenment" is generally a kind of comprehension or sentiment based on feeling and perception, and it is the most important form of human wisdom and quality development. If "play" is the external movement of hands and eyes, "enlightenment" is the internal movement of the brain. Games can provide external information for enlightenment, and enlightenment can sublimate games. If you just play and stay at the level of perception, the process of interaction between play and enlightenment is the best way to "do" mathematics.
"Enlightenment" is not only a process, but also an important result of mathematics learning. When students have "enlightenment", they really gain something. And "enlightenment" cannot be said or replaced by others, but must be an experience and epiphany under subjective efforts. Teachers can only guide themselves to experience the process of knowledge discovery and method formation through reasonable scenario creation and reasonable prototype inspiration, rather than simply telling. Nor is it a simple hint or temptation. We should take measures to fully mobilize students' thinking to "realize".