1 how to cultivate creative thinking in mathematics
Creative thinking is the unity of opposites between concentrated thinking and divergent thinking.
Centralized thinking means that people's ideas for solving problems converge in one direction, thus forming a definite answer. Divergent thinking means that when people solve problems, they start from a specific goal, think outward, think and imagine from various angles and aspects along various ways and directions, so as to explore a variety of ideas and methods to solve problems, that is, to produce a large number of unique new ideas. So many people think that creative thinking only includes divergent thinking, which is very incomplete. Creative thinking should include concentrated thinking, which is the unity of opposites between divergent thinking and concentrated thinking.
Creative thinking is the unity of opposites between logical thinking and intuitive thinking.
Logical thinking is a thinking activity that strictly follows logical laws, gradually analyzes and deduces, and finally obtains logical correct answers and conclusions. Intuitive thinking is a kind of thinking activity that relies on inspiration and epiphany to make judgments and conclusions quickly without a complete analysis process and logical procedures. Intuitive thinking can creatively discover new problems and put forward new concepts, new ideas and new theories, which is the main form of creative thinking.
Of course, logical thinking and intuitive thinking are mutually reinforcing and interrelated. Logical thinking is the basis of intuitive thinking, which is the product of highly mature logical thinking. Without the guidance of intuitive thinking, it is difficult to put forward new questions and new ideas. It can be said that intuitive thinking plays a decisive role in creative activities. However, after new ideas are put forward, it is still necessary to use logical thinking for reasoning and argumentation. Therefore, we can't exclude or belittle the role of logical thinking in creative activities. In fact, the whole development of creative thinking is carried out in the cross state of logical thinking and intuitive thinking.
2 Mathematical creative thinking and its ability training
1. Pay attention to the occurrence stage of mathematical thinking. Mathematical thinking activities can be roughly divided into mathematical occurrence stage and knowledge arrangement stage. The former refers to the process of how concepts are formed and conclusions are discovered, while the latter refers to the process of further understanding and popularizing knowledge through deduction. Therefore, the previous stage is a stage to guide students to explore knowledge and a good stage to cultivate creative thinking, so that learning and discovery can be synchronized. However, in the teaching of mathematical concept course, as long as we draw a conclusion, we are eager to put the cart before the horse and make time for practice, which is an obstacle to the cultivation of creative thinking.
2. The display of mathematical thinking mainly includes the display of three kinds of people's thinking activities, namely, the thinking activities of mathematicians, teachers and students. Teachers should build a bridge between mathematicians' thinking and students' thinking in order to realize the harmony of thinking activities. When mathematician Hilbert was teaching at the University of G? ttingen, he often raised some questions in class and solved them through discussion. His problem-solving process often benefits students a lot. Hua has always attached importance to the teaching of the thinking process of concept generation, proposition formation and idea acquisition, and to answering students' "How did you come up with it?"
3. "Question inquiry" is an effective form to carry out mathematical thinking activities. In problem-solving teaching, problem representation and problem-solving analysis are needed. In the process of thinking exploration, teachers should expose and reveal the real mathematical thinking process through appropriate methods. For example, in the face of solving a complete mathematical problem, teachers should first consider how to write such a solution, what prompted them to come up with such a solution, how I came up with it and so on. In this way, through the assistance of "process" and the revelation of the solution, the boring exercise explanation becomes vivid and concrete, so that students can know both why and why, thus gradually enhancing their innovation ability.
3 How to cultivate middle school students' mathematical thinking
Help students to set up files to answer wrong questions and cultivate their critical and comprehensive thinking.
Recording wrong examples, analyzing wrong examples and correcting wrong examples are helpful to solve the problem of "meeting but not right, being right but not complete, being whole but not beautiful" and criticize the defects of a certain thinking. Eliminate the negative influence of mindset and actively use the positive influence of mindset, thinking will be flexible rather than rigid, agile rather than rigid, profound rather than superficial, rigorous rather than omission, original rather than mechanical; Eliminating the negative influence of thinking mode and getting rid of the formal habit mode is helpful to stimulate interest and dispel doubts, and make mathematics teaching "magnetic".
Train students to think reversely and cultivate the flexibility of thinking.
For junior high school students, they are not used to reverse thinking, that is, they are not good at reverse thinking. Therefore, in mathematics teaching, it is necessary to strengthen the training of thinking, consciously guide and cultivate students' awareness and habits of reverse thinking, and help students transition from positive thinking to two-way thinking, which is conducive to cultivating students' thinking flexibility and stimulating students' interest in learning.
Strengthen divergent thinking training and cultivate students' creative thinking.
When faced with open and exploratory problems, students with rigid thinking are helpless, while students with divergent thinking are useful. The traditional students who are changeable in one question and have multiple solutions to one question also give full play to their divergent thinking. Divergent thinking is characterized by being unconventional, seeking variation, extension and diffusion, and finding various possible ways to solve problems from different angles. Strengthening the cultivation of divergent thinking will make students' thinking have a qualitative leap in the accumulation of quantity, which is conducive to creative thinking.
4 How to cultivate innovative thinking in mathematics
Guide students to learn to learn innovative thinking, and train students to learn to learn from childhood.
In teaching, students should not only learn knowledge, but also discover laws, master learning methods and cultivate innovative thinking. For example, when I teach singular and even numbers in mathematics, I ask students to name singular and even numbers within 100, write several categories and find out the rules. As a result, every student wrote some odd and even numbers cheerfully as required. Such as singular: 1 1, 13, 15, 17, 19, 1 3, 5, 7, 9, 2/kloc-. Under the guidance of the teacher, students can easily say that the odd numbers are all 1, 3, 5, 7, 9, while the even numbers are 0, 2, 4, 6, 8. On this basis, under the guidance of the teacher, the odd and even numbers within 100 we have learned have this feature, thus revealing the essence of knowledge. Students' thinking has been continuously developed, and students are interested, diligent in thinking, profound in understanding and good in memory.
Create situations, seize opportunities, stimulate and cultivate innovative thinking.
Students have a strong thirst for knowledge and high concentration. We should seize this favorable opportunity to inspire and induce them, promote their thinking and achieve good results. For example, when talking about the problem of subtracting nine from ten, I created "mother rabbit picked 13 mushrooms, and little gray rabbit took nine." How much is left? First, let the students operate the learning tools in the form of acting. After the students performed, they asked the little gray rabbit what happened after he took nine. 13 How many mushrooms are missing? How much is left? What methods are used for deletion and deletion? How to ask questions consciously aims to encourage students to explore knowledge through action and try again. When students operate learning tools, teachers patrol to guide students to fully discuss the operation process, so that students can acquire innovative thinking and gradually deepen their thinking activities.
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