1 How to Cultivate Students' Mathematical Rational Thinking
Teachers should be aware of the importance of cultivating students' rational thinking ability in high school mathematics teaching activities.
Judging from the process of psychological growth, the thinking of primary school students and junior high school students is mainly in images, and generally starts with superficial phenomena. In senior high school, students' body and mind gradually reached, and their thinking began to focus on rational thinking. They began to like to think about the essential regularity behind things. This state of mind, body and rational thinking is a great advantage of high school education. Education should fully tap this advantage, expand the rational thinking of senior high school students, and make them learn to study and live with rational thinking. Mathematics is a highly concentrated subject, so high school mathematics education is particularly important for cultivating students' rational thinking. It can be said that mathematics teachers are duty-bound to cultivate students' rational thinking ability.
Change the traditional teaching mode
Under the concept of exam-oriented education, there is a phenomenon of large capacity and fast pace in teaching. The concepts and theorems in the book have become a series of dead conclusions. Teachers pay attention to the results rather than the derivation of conceptual theorems, or there is no bone in the textbook, which leads to boring teaching. They think that the exercises in the book are of little value, but they supplement a lot of extracurricular problems, and students become "machines" to copy the blackboard. Over time, the learning spirit of students' active research has disappeared, but they always hope that teachers can provide detailed problem-solving demonstrations and get used to imitating and memorizing step by step. In this way, students can't learn the teacher's mathematical thinking and methods at all. Of course, proper problem-solving practice is a necessary condition for learning mathematical thinking, but simple problem-solving practice cannot guarantee a leap from perceptual to rational.
Lose no time to cultivate rational thinking ability in mathematics classroom teaching
Adama, a famous German mathematician, pointed out: "The process of a student exploring and solving a mathematical problem has the same nature as that of a mathematician's discovery or creation, but at most it is only a procedural difference." Therefore, in the teaching process, teachers should not only teach students the conclusion of mastering knowledge, but also reveal the process of these knowledge and conclusions, teach students the methods of exploration, and enable students to master their own ability to acquire new knowledge. Professor Zheng Yuxin pointed out in Philosophy of Mathematics Education: "Mathematics should not be equated with the collection of mathematical knowledge, but should be regarded as a creative activity of human beings". Therefore, we should seize the opportunity to cultivate students' rational thinking ability in mathematics class.
2 Mathematical thinking exercises
First of all, we should pay attention to improving students' understanding of the rigor of thinking.
Thought is the guide to action. To cultivate students' thinking rigor, we should first pay attention to the interpretation of thinking rigor and improve students' understanding of thinking rigor. Many students express problems incompletely and often make mistakes in solving problems, which is considered as "carelessness". Next time, as long as it is serious, the students' understanding is a misunderstanding. If not corrected in time, students' carelessness will never change. Teachers express incomplete problems and often solve problems for students. Be sure to analyze it clearly in time. Let the students know that the root cause of mistakes lies in their poor grasp of basic knowledge and skills and their imprecise thinking. This will improve students' understanding of the rigor of thinking and help them pay attention to the cultivation of their own rigor of thinking.
Cultivating the rigor of thinking in the teaching of concepts and definitions
Mathematical concepts and definitions are the most basic mathematical knowledge and the basis of rigorous thinking. Teachers pay attention to the teaching of mathematical concepts and definitions, that is, they should interpret concepts and definitions word by word, especially key words, and help students grasp the connotation and extension of concepts through positive and negative examples. Let students correctly understand mathematical concepts and definitions, such as absolute value concept teaching. There are two definitions of absolute value in textbooks: geometric meaning and algebraic meaning. In teaching, teachers should follow students' cognitive rules, start from examples, combine numbers and shapes, clarify the geometric meaning of absolute values, and draw the conclusion that the absolute values of any number are non-negative, so that students can understand the algebraic meaning of absolute values slowly, that is, the absolute value of positive numbers is itself, the absolute value of negative numbers is its opposite number, and the absolute value of zero is.
Cultivate students' rigorous thinking in listening and expressing.
Students' thinking is an internal process, which cannot be seen or touched. Teachers should always let students express their inner thinking process, way and rigor. When one student expresses, he asks other students to listen carefully, compare their own ideas while listening to others' opinions, and evaluate others' expressions in time. What are the advantages? What is not enough? Let the students compare and communicate with each other. Finally, the teacher will comment, praise the bright spots and analyze the reasons for the mistakes. This will help students find their own shortcomings in thinking process, way and rigor. It is helpful for students to find out where their expression is incomplete and imprecise; Help students understand things and solve problems in all aspects and links; It is helpful for students to master the thinking method of understanding things and solving problems; It is beneficial to cultivate students' rigorous thinking.
3 Mathematical thinking exercises
Strengthening variant teaching and cultivating students' rational thinking
In the process of solving "mathematical problems", it is helpful to cultivate students' ability to seek multiple solutions to one problem or multiple solutions to one problem. In problem-solving activities, students can constantly improve their thinking quality through observation, comparison, memory and imagination. By solving problems, we can cultivate their flexibility, profundity, criticism, preciseness, extensiveness and creativity, and then cultivate their calm analysis in the new situation.
Through the variant of the problem, let students learn to calmly analyze the problem and think rationally, so that they will not be anxious when they encounter problems, but look for the basis and ideas to solve the problem in an all-round and multi-angle way, so as to grasp the essence of the problem, instead of blindly touching it, leading to low quality and falling into the ocean of problems.
Enhance students' awareness of observation and association transformation.
In mathematics teaching, teachers should pay special attention to cultivating students' observation ability, which is the premise of cultivating students' comprehensive mathematics ability. They should pay special attention to those students who solve problems rashly without understanding them, let them form the habit of carefully observing the conditions and conclusions of the problems, and let them realize the effect of careful observation and careful examination of the questions through specific examples. Generally speaking, observation starts with the conditional characteristics of the problem, from observing the relationship between the known and the unknown, and from observing and analyzing the implicit relationship of the conditions.
Then comprehensively, multi-dimensionally, logically and orderly carry out association and transformation, thus solving the problem. Only by constantly analyzing and synthesizing many complex phenomena of things and rising to a theoretical understanding of the essence of things can students' learning experience leap from perceptual to rational. In-depth analysis of the form and structural characteristics of the formula in the topic, as well as the association of similarities and differences with those familiar formulas, theorems and conclusions, whether the same can be used directly, and whether the different can be used through transformation with the same structure.
4 Mathematical thinking exercises
Let students have their own time and space to think after class.
Take the students' homework now as an example. There is a lot of homework, but it has no effect. The student's goal is to finish the homework. They only care about the amount of homework completed, regardless of whether it should achieve results. In addition, some teachers always like to leave some difficult problems for students to think about, but because the problems are too difficult, some students choose to give up directly, thus failing to achieve the expected results. I have the following points about this phenomenon.
There are several methods for reference: 1. Let students take turns to give students questions, which not only solves the doubts of asking students, but also avoids the problems that are too difficult and beyond students' ability. Students will also study and look for problems that they can't because they want to leave their own problems. 2. Divide the students into groups, let them participate in the discussion in groups, and finally tell the group's solutions in class. 3. Let students choose better ideas and point out their own shortcomings. In this way, students need to ask their own questions and solve their own problems, and their thinking will open. So as to cultivate their creative thinking.
Cultivation of Associative Thinking
Associative thinking is a kind of thinking that expresses imagination, and it is a remarkable sign of divergent thinking. In the process of primary school mathematics teaching, teachers should organize colorful activities, cultivate students' associative thinking, guide students to find the connection between knowledge in imagination, exercise their abilities in flexible application, and do a good job in ensuring students' healthy development. Of course, the process of cultivating students' associative thinking is the process of letting students find the connection between abstract knowledge, and it is also the basis of students' sound development.
For example, when teaching the "trapezoid area formula" in polygon area calculation, in order to cultivate students' associative thinking, as well as to train students' learning ability and improve students' computing ability, teachers should give full play to students' imagination, organize students to divide, move and fold the trapezoid, guide students to find the connection between trapezoid and parallelogram, triangle and other knowledge, and guide students to make independent deduction.