Chapter 1 Introduction
Put forward the question of 1. 1
In the new curriculum standard of mathematics, the requirement of mathematical thinking method is further clarified: "The basic idea of the curriculum is to pay attention to improving students' mathematical thinking ability. In the process of mathematics teaching, constantly guiding students to experience the thinking processes such as observation, discovery, induction, analogy and abstract generalization is helpful for students to understand mathematical concepts more deeply and is an important embodiment of mathematical thinking ability. "Mathematical thinking method is the essence of mathematics, which is at a higher level than mathematical knowledge. It is a bridge to promote the transformation from knowledge to ability and plays an important guiding role in dealing with various mathematical problems. Paying attention to guiding students to understand mathematical thinking methods is an important guarantee to improve students' thinking ability, which can help students get rid of the sea of questions, teach them mathematical thinking and really learn meaningful things. At the same time, mathematical thinking method has penetrated into almost every special module of the new textbook, which requires teachers to study the textbook hard and conduct more teaching exploration and research on mathematical thinking method in order to promote the transformation from traditional teaching concept to modern teaching concept. I made a rough statistics on the mathematical thinking methods involved in the mathematics textbooks for primary and secondary schools compiled nationwide. From the statistical results, we can generally see several words with high frequency: induction, abstract generalization, inductive conjecture, combination of numbers and shapes, etc. Generally speaking, deduction has been paid enough attention in the whole mathematics teaching, while induction, as a method to cultivate students' creativity, has been neglected. This contrast is thought-provoking. The author thinks that this reflects all kinds of disadvantages in current mathematics teaching from one side, that is, when introducing new concepts, only the definition is expounded and the formation process of concepts is ignored, and when explaining exercises, only the steps of solving problems are explained without paying attention to exploring the ideas of solving problems.
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1.2 Research Summary at Home and Abroad
Status quo of foreign research: Since the 20th century, due to the formation of axiomatic mathematics and the in-depth study of the basic theory of mathematics, people have gradually paid attention to the internal relations between the branches of mathematics, and paid more attention to the emergence and development of mathematical thinking methods. Many famous mathematicians abroad have made theoretical research on mathematical thinking methods, and have also achieved rich research results. The Hungarian mathematician Paulia wrote "Mathematics and Conjecture" and other classic works. His main point is that there are two kinds of reasoning in mathematics: argumentation reasoning and rational reasoning. He revealed the internal relationship between them, that is, they belong to two aspects and two forms of thinking, and they interact with each other in the process of mathematical discovery and creation. Mathematics thesis should not only attach importance to the application of argumentation reasoning, but also attach importance to the study of perceptual reasoning, which can enrich our scientific thinking and improve our innovative ability. And mathematical inductive reasoning is a special case of rational reasoning referred to by Paulia. In addition, there is the Spirit, Thought and Method of Mathematics collected by Mishan National Library, which brilliantly discusses the spiritual essence of the whole mathematics and the important mathematical thoughts running through it, and also provides some good references for the teaching of mathematical thinking methods. He believes that in primary and secondary schools, we should pay attention to cultivating students' ability to solve mathematical problems in real life by using mathematical thinking methods. At the same time, the author summarizes and refines the more common and valuable mathematical ideas in mathematics from the perspective of mathematical development.
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The second chapter is related theoretical research.
2. 1 Definition of related concepts
2. 1. 1 the connotation of mathematical thinking method
Before expounding the definition of mathematical inductive thinking, we should first make clear the connotation of mathematical thinking method. The so-called mathematical thought is a mathematical viewpoint that is highly abstracted from concrete mathematical content, and it is the guiding ideology for solving problems by using mathematics. In fact, it is an understanding of the essence of mathematics. Mathematical method refers to the comprehensive strategy for people to analyze and solve mathematical problems, that is, the format and steps of solving problems, which is an effective means to implement mathematical ideas. Mathematical thought and mathematical method are interrelated, but they are different. Mathematical thought is the spiritual essence of mathematical method, and mathematical method is the external expression of mathematical thought. In other words, mathematical thinking is implicit, while mathematical methods are explicit. At the same time, the characteristics of mathematical thought are universality and generality, and the characteristics of mathematical methods are concreteness and operability. Mathematical thought is the sublimation of mathematical methods, which can reflect the internal relationship between mathematical contents more profoundly than mathematical methods. However, they are all carriers of thinking activities. In the process of using mathematical methods to solve problems, when perceptual knowledge accumulates to a certain extent, it will rise to mathematical thought, and the formation of mathematical thought can play a certain guiding role in mathematical methods. So it is not easy to distinguish them strictly. In Selected Lectures on Mathematical Methodology, Mr. Xu Lizhi did not give a clear definition of mathematical thought, mathematical method and mathematical thinking method, and hardly distinguished them when using them. Although Mr. Zhang Dianzhou explained the mathematical method and the mathematical thought respectively in the Draft of Mathematical Methodology, he also thought it unnecessary to distinguish them deliberately. He generally called them 17. (model essay. )
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2.2 Research based on relevant theoretical basis
Absorbing all kinds of valuable ideas from philosophical schools can provide guidance for our more meaningful teaching of mathematical inductive thinking. 2.2. 1 theoretical basis of educational psychology
1. bruner's cognitive theory: discovery learning theory
In 1960s, Bruner first raised the issue of transfer in education, which aroused widespread concern. He advocates "discovery learning", which can stimulate students' potential and promote their discovery and creation. He believes that there is a common phenomenon in learning, that is, learning can be transferred. If students' ideas are at a higher abstract level in the cognitive structure, it will be more conducive to students' learning. Judd, a famous American psychologist, also conducted migration experiments. The results show that mastering the general principle is beneficial to the transfer of learning, and the general principle in mathematics is mathematical thinking method. From this theory, we can see that the process of mathematical induction is actually a process of discovery learning, and it is also very beneficial to guide students to consciously use mathematical induction to realize learning transfer, which can quickly improve their mathematical thinking ability. At the same time, paying attention to knowledge analogy in teaching is helpful to improve students' inductive ability.
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Chapter III Investigation on the Teaching Situation of Mathematical Induction Thought ....................... 14
3. 1 The present situation of inductive thinking teaching in mathematics and teachers' confusion ............................. 14
3.2 Students' mastery of the content of mathematical induction .............................. 16
The fourth chapter is the arrangement and analysis of mathematical induction in primary and secondary school textbooks. ..............................2 1.
4. 1 The penetration point of mathematical induction in primary school textbooks ................................ 22
4.2 ........................, the penetration point of mathematical induction in junior high school textbooks, 27
4.3 ....................., the penetration point of mathematical induction in high school textbooks, 3 1
4.4 Comparative Analysis of Mathematics Inductive Thinking Courses in Primary School, Junior High School and Senior High School ................... 36
The fifth chapter explores the teaching of mathematical inductive thinking under the background of the new curriculum.
5. 1 Basic principles of inductive thinking teaching in mathematics
Bruner, a famous American psychologist, emphasized in the Basic Theory of Discipline that "understanding the basic principles and principles to be followed will make the discipline easier to understand" and "understanding the basic principles can narrow the gap between' primary' knowledge and' advanced' knowledge." Therefore, the following basic principles should be followed in the teaching of mathematical induction.
(1) heuristic principle
This principle aims to emphasize the subjectivity of students in the learning process, and the role of teachers is to inspire and guide students and mobilize their learning enthusiasm. This requires teachers to fully consider students' existing knowledge and experience, carefully create problem situations in students' "zone of proximal development", stimulate students' curiosity in learning, guide students to learn to question, observe and think spontaneously, and consciously abstract and summarize the essential attributes of knowledge in mathematics test papers. At the same time, it is also important for teachers to grasp the appropriate guiding opportunity in teaching. Confucius, a great educator, famously said, "If you don't get angry, you will gain power." It means that students should be inspired when they think but can't figure it out, that is, they must be given a thinking process before inspiration, so that students can think positively first and then get inspiration in time.
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Chapter VI Research conclusions and suggestions.
6. 1 Research conclusions and suggestions on curriculum setting
After careful analysis of mathematics textbooks of all grades, the author thinks that there are such problems: the textbooks of inductive thinking course content do not give teachers a clear instruction, and the content is not clear enough. At the same time, the arrangement of the teaching content of infiltrating mathematical inductive thinking is not systematic and generally scattered. Curriculum standards only require teachers to pay attention to cultivating students' inductive thinking ability on a macro level, and there is no evaluation standard. Therefore, in view of these problems, the author gives the following suggestions:
One is to set up a small chapter in each grade textbook to introduce inductive reasoning, and penetrate into the knowledge points of each chapter, so that it can be subdivided into specific goals to achieve. In my opinion, the exercise configuration of inductive thinking after algebraic expressions in the seventh grade textbook of People's Education Press is reasonable and should be followed. Because students have just learned algebraic expressions, they have learned to use letters instead of ordinary numbers in the format of mathematical papers, so that students can slowly transition from concrete thinking to abstract thinking, which is very important for completing the induction process. This setting can just provide a foundation for completing the practice of inductive thinking, which can consolidate the study of algebraic expressions.
Secondly, in order to improve students' understanding of the importance of inductive thinking, the reading materials in the textbook can set up more famous conjectures, which can also stimulate students' enthusiasm for exploration. For example, when learning prime numbers in the fifth grade, you can add a reading link at the back of the content-Goldbach conjecture, one of the three major conjectures in the world. The four-color conjecture is also an interesting conjecture, so it is helpful for students to understand it and improve their interest in learning. Thirdly, because the development of students' thinking in all grades is orderly and hierarchical, the course content of inductive thinking should be coherent and progressive from primary school to junior high school to senior high school, following the development principle of from shallow to deep and from low to high, and paying attention to the different needs of students at different levels.
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References (omitted)