Abstract: Mathematics should not be regarded as a simple tool, but it is also of great significance to thinking training. College students should cultivate the visualization, abstraction, intuition and functional thinking of mathematics. To cultivate college students' mathematical thinking, it is necessary to optimize the way of thinking and cultivate their logical thinking ability and intuitive thinking ability.
Keywords: mathematics; College students; elaborative faculty
First, the concept and structure analysis of mathematical thinking
Mathematical thinking, as a special form of thinking, is a reflection process of indirect generalization of mathematical objects by human brain using mathematical symbols and mathematical language. Specifically, mathematical thinking is a cognitive process that takes mathematical concepts as cells and reveals the essence and internal relations of mathematical objects through mathematical judgment and mathematical reasoning. Mathematical thinking is not only subject to the general thinking of human beings, but also restricted by the laws of general thinking, and has characteristics different from general thinking. Mathematical thinking is a kind of advanced thinking and belongs to the category of modern abstract thinking. The functional structure of mathematical thinking is a three-dimensional structure, and the three axes are thinking content, thinking method and individual development level. The interaction of these three parts constitutes mathematical thinking ability. Mathematical thinking ability is the core of all kinds of mathematical abilities, and the content is the thinking object faced by the thinking subject, including mathematical concepts, laws, propositions, various mathematical theories and practical problems. Mathematical thinking method is the core of mathematical method, the step and form of mathematical thinking activity, and the way and procedure to deal with thinking content. The level of individual development refers to the thinking quality and non-intellectual quality of the subject, in which the thinking quality includes profundity, extensiveness and flexibility, and the non-intellectual quality includes motivation, emotion and will that play an important role in thinking activities.
Second, what kind of mathematical thinking ability should be cultivated?
(1) thinking in images. Image thinking is concrete thinking, including non-operational forms (observation, perception, etc. ) and operation form (direct operation on things or their models, etc. ). College students have greatly improved their senses and operations, and their abilities have been enhanced to a certain extent. Their memory style has gradually changed from mechanical memory to understanding memory, and they are eager to learn independently.
(2) abstract thinking. Abstract thinking is a thinking activity closely related to abstract activities, and it is the core and foundation of higher mathematics. Abstract thinking fully embodies the high rigor and rigor of higher mathematics, and it is also the mathematical thinking that students need to pay attention to. The abstraction here has duality, that is, while extracting its essential attributes, it strips off other non-essential attributes.
(3) Intuitive thinking. Intuitive thinking is a special understanding method. It is a way of thinking with keen imagination and quick judgment on mathematical objects, structures and laws, which is characterized by directly solving problems or getting the truth.
(4) Functional thinking. Functional thinking refers to a kind of thinking to understand things from the relationship between mathematical objects and properties. Function is a key research object in higher mathematics, and many problems we solve in real life involve the judgment and solution of function relationship.
Thirdly, how to cultivate college students' mathematical thinking ability.
It is a long and arduous process to cultivate college students' good mathematical thinking. The basic strategy is to emphasize the formation of ideas and promote the cultivation of ideas. Please pay special attention to the following points:
(1) Optimize the way of thinking. If the students' understanding of what they have learned is not profound and accurate, or they cannot establish the connection between old and new knowledge, it will lead to insufficient understanding and deviation, and their thinking is not rigorous and flexible enough when solving specific problems. Therefore, it is necessary to guide students to optimize their way of thinking and cultivate the rigor and flexibility of thinking.
1. Correct the mistakes in thinking and cultivate the rigor of thinking.
Some students do not pay attention to the hidden conditions in the problems they have learned when solving mathematical problems, which leads to wrong thinking and affects the correct solution of the problems. Therefore, students should be taught to fully tap the implicit conditions, adjust the thinking process in time, correct the thinking mistakes and cultivate the rigor of thinking.
2. Change the angle of thinking and cultivate the flexibility of thinking. When solving problems, students are used to deducing conclusions from the known, forming one-way thinking, which brings certain thinking obstacles to solving problems. The cultivation of reverse thinking should run through the whole learning process.
3. Cultivate and develop students' mathematical inquiry ability, and then stimulate students' innovative thinking. The exploration and innovation ability of mathematics is the most creative and challenging element in mathematical thinking, and it is also the core of mathematical thinking. The history of mathematics development for thousands of years is a history of continuous exploration and innovation.
(2) Cultivate the ability of logical thinking. Logical thinking ability is an important part of thinking ability. The main forms of logical thinking are concepts, judgments and reasoning, and they are the main tools to prove conclusions. Logical thinking is used for abstract definition, formula derivation, theorem proving and problem solving with knowledge.
1, cultivate the ability to understand concepts and apply concepts to solve problems. Understanding ability is the basis of learning mathematics. If students don't have a deep understanding of the occurrence and development of some mathematical concepts or principles, they can't grasp the essence of the problem. Therefore, we should deeply understand the essence of concepts, laws, formulas and theorems and apply concepts to solve problems.
2. Cultivate the ability of reasoning and judgment. Reasoning and judgment ability is an important part of logical thinking ability. Cultivating the ability of reasoning and judgment should be based on students' deep understanding of concepts. Students should master the necessary reasoning and judgment methods such as induction, deduction, analogy, exhaustion, special cases and reduction to absurdity, and consolidate them through certain training, so as to improve their reasoning and judgment ability. To improve students' reasoning ability, we should pay attention to the learning of reasoning process (including logical reasoning and intuitive reasoning), and form the habit and gradual strictness of reasoning process from the beginning.
3. Cultivate students' abstract generalization ability. We should be good at abstracting the number-shape relations reflected in mathematical materials from concrete materials and summarizing them into concrete general relations and structures, and do a good job in the demonstration of abstract generalization, paying special attention to analysis and comprehensive learning; In addition, when solving problems, we should pay attention to exploring the universality hidden behind various special details, find out its internal essence, and be good at grasping the main, basic and general things; At ordinary times, students should be encouraged to summarize some problems regularly and cultivate the habit of induction.