1 How to master mathematical thinking
Examples are cited to form mathematical representations, and essential features are summarized to produce mathematical concepts.
The quantity, quality and given time of specific cases directly affect students to form clear representations, which is the key for students to establish correct concepts. Therefore, first of all, we should choose standard examples to provide students, so as to present the essential attributes of concepts to students correctly, directly, clearly and vividly, and form a clear representation as the basis for students to form concepts. Secondly, case analysis is a logical treatment of cases, which makes the concept concrete through comparison, analogy, induction and abstraction of the same essence of things. When students have a preliminary correct understanding of the concept and a deeper understanding of the essential characteristics, in order to make the connotation and extension of the concept clearer, some positive and negative examples can be appropriately selected for analysis, thus highlighting the essential attributes of the concept.
Through activities such as variant observation, it is beneficial to cultivate students' habit of looking at problems comprehensively. However, variant examples should not be given too much or too early, which requires teachers to carefully consider and avoid randomness. You can't pretend to be the master and interfere with the formation of a clear representation.
Problems arise in the process of solving problems, thus generating mathematical concepts.
The teaching process is a continuous activity of asking questions and solving problems, so teachers can ask some questions with appropriate difficulty, guide students to think actively and explore independently, find and ask questions in analysis and reasoning, and teachers can introduce mathematical concepts in time.
In this way, students not only clarify the significance of concept introduction, but also strengthen the important position of mathematical concepts in the process of solving problems. In this process, we can give full play to students' subjective initiative, guide students to think positively, make bold guesses and describe accurately, which will help students to deeply understand the essence of concepts, lay a good foundation for the expansion and flexible application of concepts, and cultivate students' profound thinking.
2 mathematical thinking methods
Closely follow the essence of concepts, promote the series and integration of concepts, and form a three-dimensional network of concepts.
Through the extensive and close contact between old and new knowledge, the abstract thinking mode of mathematics is revealed, the capacity of knowledge is expanded, the concept is further consolidated and deepened, and the flexible application ability of knowledge is increased, which is conducive to the formation of structured and systematic concepts of mathematics. The related concepts are combined to form a knowledge network system, and the concepts acquired by students are accumulated layer by layer. Teachers should be good at guiding them to connect relevant knowledge vertically and horizontally, so that students can outline a three-dimensional concept network from a certain concept point and form an overall understanding. For example, in the teaching of junior middle school function, the concept of function is gradually formed through the understanding of the changing relationship of quantity in life, and then the linear function, inverse proportional function and quadratic function are merged together. After fully grasping the essential characteristics of each function, the differences and connections between them are analyzed and summarized to deepen the understanding of the concept of function.
Some concepts in mathematics are interrelated, influenced and interdependent. We should be good at guiding students to contact related concepts in time, fully revealing their inherent laws, so that students can have a comprehensive and systematic understanding of the concepts they have learned, help students analyze mathematical problems when solving problems, and accurately locate the mathematical concepts to be used.
Strengthen the practical application of concepts, deepen the understanding of the essence of concepts and improve the mathematical thinking ability.
The process of concept formation is the process of understanding the concept, and the process of applying and consolidating the concept is the process of further remembering and maintaining the concept. In order to apply abstract concepts to solving specific problems, we can deepen our understanding of concepts through activities such as identification, judgment, reasoning and operation, so as to achieve a higher level of application.
Students define the concept, but also need to strengthen the consolidation of the concept through a certain amount of application training, deepen the understanding of the concept, and make their own concepts more systematic and skilled, which requires teachers to train students in a planned and hierarchical manner. Teachers should carefully select design examples and exercises to further highlight the application of concepts. The choice of topics should be targeted and varied, such as multiple-choice questions, fill-in-the-blank questions, or comprehensive questions, in order to strengthen the concept. It is also necessary to purposefully design some problems with hidden conditions or set some interference factors for the places that are easy to make mistakes in mathematical concepts, so that students can enhance their understanding and application ability of concepts in discrimination. For example, to understand the definition of quadratic function, we can design the following exercises: If the function y = (m-3) xm2-3m+2+(m+1) x-2 is a quadratic function, find the value of m; Set the simplification problem of quadratic root:-1a, stick to the concept and grasp the implicit condition A < 0.
3 Mathematical thinking method
According to the knowledge points in the textbook, cultivate students' language expression ability.
The cultivation of students' language expression ability is not only the teaching task of Chinese subject, but also the cultivation of students' language expression ability in mathematics class according to the knowledge points of textbooks. This kind of teaching is conducive to cultivating students' ability to analyze and solve problems. For example, when teaching the content of "location" in the second volume of the national standard experimental textbook for grade one, I first let the students observe the school supplies on the desk, and then let the students observe the theme map after expressing the upper and lower objects in words, so that the students can accurately describe who is above whom and who is below whom in clear language. Then guide the students to use the resources in the classroom and describe them accurately with "up, down, front, back, left and right". When a student speaks with these directional words, he says, "The teacher is above the podium and we are below it. Rebecca is in front of me, followed by Li Fang, Zhao Wei on the left and Zhang Hang on the right. " This kind of training not only trains students' ability to distinguish "position", but also trains students' language expression ability, which lays a solid foundation for future study and development.
Strengthen divergent thinking training and broaden students' innovative horizons
High school students often put forward their own views on some issues. This psychology of seeking difference and seeking knowledge is guided in mathematics, which often shows divergent thinking. Therefore, we should pay more attention to the reasonable factors in students' thinking and encourage "innovation". In teaching, teachers should adopt various means, such as inspiration, practical activities and multimedia demonstrations. Guide them to open up their minds, broaden their thinking, analyze and solve problems from different angles, which is conducive to the training of innovative thinking.
For example, to find the maximum and minimum values of the function f(x)=sinθ-cosθ-2, we can use the following ideas: (1) Use the boundedness of trigonometric functions to solve the problem; (2) Using variable substitution, it is transformed into a rational fractional function to solve; (3) Using the slope formula in analytic geometry and transforming it into the geometric meaning of the figure to solve it, and so on. Through this question, students are guided to seek answers from trigonometric function, fractional function, analytic geometry and other angles. , communicate the connection between knowledge, overcome the mindset, broaden the breadth of innovation, thus cultivating students' divergent thinking ability.
4 Mathematical thinking method
Make good use of thematic maps to stimulate students' interest in learning.
Make good use of theme maps to stimulate students' interest in learning. Illustrated pictures and texts are a major feature of the first new textbook, and the arrangement of the theme map of the textbook fully reflects that mathematical knowledge comes from life and moves towards life. For example, when learning "length unit", let students use their own tools to measure according to the tips of the theme map, so that students can have cognitive conflicts and stimulate their interest in learning. Every theme map is a representation of life. "The extension of the classroom is life". If we closely link the classroom with students' lives, students' learning will be full of endless fun.
Use students' curiosity to stimulate their interest in learning.
As the saying goes, interest is the best teacher. In the process of primary school mathematics teaching activities, we can make full use of students' curiosity and cultivate their interest in learning mathematics. Curiosity refers to people's psychological and behavioral tendency to explore new things, and it is the internal driving force to realize the process of creative thinking. At the same time, when curiosity turns into curiosity, it will produce rich imaginative thinking, which will help students improve their mathematical ability. For example, when explaining the inner angle of a triangle and this knowledge point.
We can ask students to prepare a triangle in advance and let them measure the degree of each inner angle and record it. Then we can invite a student to quote the degree of any two internal angles of the triangle he has measured at will, and the teacher can answer another degree accurately. At the beginning, students are bound to have doubts and strong curiosity. "How on earth did the teacher know the degree of another angle in such a short time?" Only in this way can we effectively attract students' attention and help them cultivate mathematical thinking and good study habits.