The author puts forward his own views on cultivating senior high school students' mathematical thinking ability.
First, cultivate students' habit of thinking boldly in teaching.
Everyone should learn to study. Only by thinking boldly in the process of continuous learning can we acquire new knowledge, renew our concepts and form our own new understanding. In the history of mathematics, Descartes, a great French mathematician, liked reading extensively when he was a student. He realized the disadvantages of the separation of algebra and geometry, boldly tried to study geometric drawing by algebraic method, and pointed out the relationship between drawing and solving equations. Through specific problems, the coordinate method is proposed to express geometric curves as algebraic equations. It is asserted that the degree of curve equation has nothing to do with the choice of coordinate axis, and curves are classified by the degree of equation to understand the relationship between the intersection point of curves and the solution of equations. It advocates the combination of algebra and geometry, and applies quantitative methods to geometry research, thus creating analytic geometry. These achievements are inseparable from Descartes' bold thinking. Therefore, high school math teachers should not only let students learn, but more importantly, guide students to think boldly in the process of active learning, constantly think actively in learning, know the background of knowledge generation in thinking and experience the process of knowledge generation. Therefore, we should fully show the process of knowledge generation and development in the teaching process, so that students can experience a "re-creation process", actively participate in the practice of understanding things, and understand the spirit and thinking methods of mathematics.
Second, guide students to think vertically and deepen their learning at different levels.
When learning new knowledge in each chapter, teachers should guide students to think vertically, change conditions and actively explore, so as to learn these new knowledge in depth.
For example, in the course of compulsory 4 of People's Education Press, learn the basic relationship of trigonometric functions with the same angle. In order to make students skilled and consolidate these two formulas, we can design an example: known sin? Cut =0.6, cut into the second quadrant angle, and find cos? Zhuo, Tan? Stop. After the students solve this problem with the basic relationship of trigonometric function and the same angle, the teacher can change the conditions and change this problem into a known cos? Cut =-0.8, cut into the second quadrant angle to find sin? Zhuo, Tan? Let the students solve this variant problem independently. After the students analyze this variant problem with two formulas, the teacher can continue to guide the students to think: "Can you change one condition in the problem to get other variant problems and find the corresponding solution?" Teachers can let students discuss in study groups on the basis of positive thinking. Students produce the following variants through discussion: variant ①, known crime? Cut =0.6, cut into the first quadrant angle, and find cos? Zhuo, Tan? Cut; Variant 2, known as crime? Cut =0.6, find cos? Zhuo, Tan? Cut; Variant ③, commonly known as cos? Cut =-0.8, looking for guilt? Zhuo, Tan? Cut; Variant 4, commonly known as Tan? Cut =-0.75, looking for guilt? Hey, because? Cut and wait. In this way, students are more interested in studying the variant problems that students actively think and explore, and more confident in completing their own solutions, which will play a multiplier role in skillfully using the basic relationship between the two trigonometric functions in this section. Teachers guide students to think vertically in the teaching of "new teaching", students' initiative to explore knowledge is enhanced, and the learning effect is naturally enhanced.
Third, guide students to think horizontally and deepen the connection between knowledge through more thinking.
Teachers can guide students to think horizontally in the review class, which can help students to contact the knowledge of each chapter they have learned, find the combination point between knowledge and form a strong knowledge network. Teachers can guide students to think and learn from multiple angles, levels and in all directions, so that review can be deeper and more effective.
For example, when reviewing the knowledge of solving triangles, teachers can choose the following examples: in △ABC, the bisector AD of ∠ A = 60 divides BC into two sections, and the ratio BD: DC = 2: 1, AD=4■, (1) finds the length of the triangle; 2 find the angle C.
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Teachers can guide students' thinking in this way. This problem is a problem of solving triangles. We have studied triangles, including sine theorem, triangle area formula, cosine theorem, trigonometric function and some properties of triangles. Let students think more about whether they can solve this problem from different angles.
Many students have other solutions. The whole classroom atmosphere is active, and students explore knowledge actively rather than passively. In the discussion and cooperation, they develop lateral thinking, deepen the impression of understanding this part of triangle knowledge, cultivate the broadness of thinking and improve the ability of divergent thinking.
Fourth, constantly stimulate students' thinking process and enhance good thinking motivation.
Students need to constantly enhance the motivation of thinking activities in order to keep their thinking activities in a good state. In teaching, teachers should affirm students' thinking design, give more incentives at critical moments, and enhance students' thinking motivation. Teachers should pay attention to evaluating students' thinking achievements in time in teaching, even if there is a little success, they should encourage and praise them in time to enhance students' thinking enthusiasm; Perfect supplement to students' imperfect thinking through collective strength, so that they can feel the joy of success; Correct and encourage students' mistakes in the process of thinking in time, constantly enhance their self-confidence, help them find out the reasons for their failure, and let them raise the sails of thinking.
Fifth, inspire students to have direction and purpose in the process of thinking.
In the process of high school mathematics teaching, teachers should grasp the students' thinking direction, so that students can start from a certain aspect or angle of the problem, and the students' thinking range will not be too large, so that students' thinking will be more concrete and feasible. Teachers should let students know what problems to solve and what goals to achieve in the teaching process, which is more conducive to students' thinking. For difficult problems, teachers should give hints or examples to open the door of students' thinking. Teachers should not only presuppose and restrict students' thinking, but also restrict it too much, so as not to bind students' thinking and hinder their progress. In order to achieve the best effect of students' thinking, teachers need to make more presuppositions of students' thinking before class, know more about students' actual level, and make teaching closer to students' thinking level, so as to obtain better results.
In short, the formation of students' thinking ability needs a long-term systematic training process. It may be time-consuming and laborious for teachers to carry out students' thinking activities, and sometimes they may not be able to complete the so-called "learning tasks" as planned, but students will gain more if they continue to actively train students' thinking. As long as teachers and students attach importance to this point and scientifically train their thinking in a planned, purposeful and step-by-step way, more outstanding talents will be cultivated.