1 How to develop children's mathematical thinking
"Wake up when you are called" in classroom teaching
In normal teaching, teachers should not just tell stories, but should pursue the awakening and activation of children's experience and thinking. This should be the pursuit of mathematics teaching. Only when children's intrinsic motivation is awakened, and only when children's subjective thinking is activated, can effective mathematics learning take place. Years of teaching practice have confirmed my simple intuition again and again, which has enabled me to accumulate more and think rationally about this issue in practice. Especially how to skillfully stimulate students' cognitive conflicts? How to use counterexamples consciously? How to properly use teaching skills such as reduction to absurdity and "playing dumb" in the teaching process? Wait a minute. These will actively awaken children's inner learning motivation and enthusiasm, and let children participate in the process of mathematics learning more consciously and actively. For example, after students initially perceive the concept of triangle, they ask, "What kind of figure is a triangle?" At this time, students may say several answers: ① surrounded by three lines, ② composed of three lines, ③ spelled by three lines, and so on. Here, we need the teacher's clever guidance. How did you spell these three lines just now? End-to-end connection is called "encirclement". Now who can give a complete definition of triangle? The concept of "a figure surrounded by three lines is called a triangle" has formed a clear representation in students' minds and also enlivened the classroom atmosphere.
For example, in the class of teaching area and area unit, when students initially establish the representation and concept of "square centimeter" through mathematical activities such as looking, touching and thinking, and then measure the area of some object surfaces or plane graphics with the "square centimeter" model in their hands, I said without trace, "Now, please measure the area of the desktop with the square centimeter model in their hands." As soon as the problem came out, some students really began to measure. More students first looked at each other and then exploded quickly: "Teacher, the class table is so big, when will it be measured?" "The square centimeter is too small to measure the area of the desk!" "Teacher, is there an area unit larger than square centimeters?" Undoubtedly, the existing area unit is too small and the area to be measured is large, and there are strong contradictions and conflicts between the new task and the existing knowledge. And this kind of conflict just contains the infinite possibility and space for students to move towards new knowledge. Isn't students' mathematics learning awakened and activated again and again in such contradictions and conflicts?
"Dance with the Drum" in Classroom Teaching
As the saying goes, failure is the mother of success. Indeed, necessary setbacks, coupled with effective reflection on failure, may help individuals get rid of the shadow of failure and achieve a leap from failure to success. However, for children who are still in the process of physical and mental development, I prefer to believe the following judgment, that is, "success can feedback success more." Mathematics is undoubtedly abstract, but children's thinking is still in a period of mainly thinking in images and gradually transitioning to abstract thinking. The contradiction and opposition between the abstraction of content and the visualization of thinking undoubtedly made many students form a negative impression or fear of mathematics from the beginning. At this time, teachers should not only use the necessary teaching methods to resolve the abstraction of mathematics itself, but also stimulate their interest and enthusiasm in mathematics learning through positive comments such as encouragement, affirmation and appreciation, and encourage them to make continuous progress on the road of mathematics learning, so that they can discover the value of mathematics learning and gain a successful experience in mathematics learning.
For example, in the teaching of "area and area unit", I am always glad to be encouraged and spurred by students in the teaching process: "Good idea!" "I really think about the problem!" "The area unit you created is the same as that created by mathematicians!" It is conceivable that if our students are always affirmed, respected and appreciated in the process of mathematics teaching, their enthusiasm and creativity will undoubtedly be better stimulated and awakened. This is the power of inspiration.
2 How to cultivate students' mathematical thinking
Optimize classroom teaching and open divergent thinking
Classroom teaching is the main position and battlefield to implement quality education, and it is also necessary to start with classroom teaching to cultivate students' divergent thinking ability. The cultivation of thinking ability is one of the core goals of mathematics classroom teaching. In classroom teaching, teachers are like directors and students are actors. Therefore, teachers should organize teaching in various ways to realize diversification of teaching objectives, scientification of teaching contents, optimization of teaching methods and multidirectional information transmission, guide students to put forward new ideas, schemes and methods to solve problems in time, create a positive and harmonious teaching environment, and open the door for students to divergent thinking.
Try to deal with teaching materials flexibly. Stimulate the imagination of the brain by changing the angle and conditions of thinking. For example, after teaching that a balance is a tool for measuring the mass of an object, the teacher can ask: How can you measure the mass of an object without a balance? At this time, it is necessary to broaden the thinking space according to the students' answers and the situation, so as to improve learning efficiency and cultivate divergent thinking ability.
In terms of teaching methods, teachers try their best to use teaching methods such as watching, listening, reading, thinking and practicing to keep students' brains in a positive and exciting state and create favorable conditions for students' thinking development. For another example, teaching can be assisted by practical activities and knowledge contests, which can stimulate and induce students to open their minds, tap their potential, truly realize the coordinated use of eyes, ears, mouth and brain, and achieve the purpose of cultivating students' divergent thinking.
Diversification to cultivate students' thinking quality
(A) the use of open questions to cultivate students' profound, broad and creative thinking.
First of all, the conclusions of open questions are different or the problem-solving strategies are diversified, but there is an internal connection between these different conclusions or diversified problem-solving strategies, that is, "the form is scattered but the spirit is not scattered." [Case] When I was talking about the section of the vertical diameter theorem, I designed such a set of questions: (1) In ⊙O, the chord AB=8cm, the distance from point O to chord AB is 3cm, and the radius is. (2) For 5cm, chord AB=8cm, find the distance from O to chord AB. (3) If the radius ⊙O is 5cm and OP=3cm, what is the shortest chord length among the chords passing through point P? (4) If P is the midpoint of the arc AB, the distance from P is 2cm, and the chord AB=8cm, find the radius ⊙ O ... "
Students themselves get the auxiliary line of this kind of problem through practice: Rt△, whose three sides are half of chord length, radius and chord center distance. So as to effectively cultivate the profundity of students' thinking. Secondly, students are also broad in solving problems, that is, they do not use the knowledge learned in this unit or this textbook to solve problems.
(B) The use of conjecture is a means to cultivate students' creative thinking.
Regarding conjecture, Paulia has a wonderful exposition: "I want to make a small suggestion. Can students guess the result or part of the result of the question before doing it?" . Once the student expresses his basic idea, he connects himself with the question and is eager to know whether his guess is correct. Therefore, he will take the initiative to care about this problem, care about the progress of the class, and will not take a nap or make small moves. "From Paulia's exposition, we can feel that students don't need to have guesses like scientists, but all guesses that can promote students' learning and cultivate students' creative thinking are very meaningful. Guide students to guess, let students acquire knowledge better, show their innovative ability and improve their self-confidence in learning.
3 How to cultivate students' thinking ability
Renew ideas, create a teaching environment, and encourage diversified independent thinking methods.
Mathematics classroom is no longer simply regarded as a place for students to "accept" knowledge, but should be a place for students to explore and exchange mathematics and establish their own effective mathematical understanding. Teachers should strive to create a teaching environment in which students are good at thinking and willing to learn, so that students can form a correct learning style and attitude towards mathematics in the process of classroom learning, pay full attention to students' emotional input in mathematics learning, make students feel happy and enriched, and let students take the initiative to learn, participate in thinking activities and experience a process of practice and innovation.
[Case 2] Teaching "Prime Numbers and Composite Numbers". 1, create a situation. Teacher: How about being a boy scout today? Courseware display: deciphering the password-in an operation, our scouts seized the enemy's password, and the first number was the largest prime number within 10; The second number is about 3, which is also a multiple of 6; The third number is neither prime nor composite; The fourth number is both prime and even; The fifth number is a composite number, which is an odd number within 10. Who can crack the password? This introduction aroused students' strong desire to apply mathematical knowledge to explore and solve practical problems. The new example is 1. Teacher: According to the approximate number of each number, the numbers from 1 to 12 can be divided into several situations. Divide one point by yourself, then communicate in groups, find three groups to report and fill in the results on the blackboard respectively. Write one divisor, two divisors and more than two divisors. Teacher: So what problems are we going to solve in this class? Let the students read by themselves and see what you can learn from the textbook. Students read books and learn to understand the concepts of prime numbers and composite numbers by themselves. Finally, exchange the concepts of prime number and composite number in class. Teacher: The courseware shows the concepts of prime number and composite number. 3. help decipher the password. In this process, through group discussion, teachers give guidance, find out the relationship between them, encourage students to actively participate in mathematics activities, give full play to students' main role, and enable students to actively explore and acquire knowledge. At the same time, let students feel the happiness of learning, experience the happiness of success and inspire their thinking.
Renew teaching concepts, improve teaching methods, stimulate motivation and cultivate students' thinking consciousness and quality.
Teaching should not only improve students' basic mathematics literacy, but also cultivate students' innovative consciousness and practical ability to promote their all-round development. At present, it is an urgent task to cultivate primary school students' innovative thinking, innovative consciousness and practical ability, and the premise is to stimulate motivation. Psychologist Bruner regards "motivation principle" as an important teaching principle, and thinks that teaching must stimulate students' learning enthusiasm and initiative. Interest can generate learning motivation, and with interest, teaching can achieve good results.
[Case 1] For example, in the teaching of "Meeting Problem", in order to clear up the learning obstacles, classes are started and a situation is created: first, two students walk face to face from both ends of the classroom and ask, "What are the directions of these two students?" "What is the result of walking?" Through the intuitive demonstration of real life, students' emotional understanding is enriched, so that students can correctly understand abstract concepts such as "relativity", "encounter", "separation" and "coexistence" and actively participate in the exploration of new knowledge. Then through the teaching methods of discovery, heuristic and discussion, the initiative and consciousness of students' thinking are mobilized.
4 How to cultivate students' reverse thinking
Induce reverse thinking
In classroom teaching, teachers should link old and new knowledge according to the knowledge they teach and students' cognitive laws, change their angles, seek transformation and variation, and induce students' reverse thinking. For example, after teaching an application problem, teachers can change the conditions and problems in the problem to make it another application problem, and guide students to analyze and think in the opposite direction. The advantage of doing this is not only to cultivate the flexibility of students' thinking, but also to effectively guide students to consolidate their knowledge. At the same time, it can further promote students' understanding ability, gradually guide students' thinking to a new situation and cultivate students' thinking of seeking the opposite sex.
In the teaching of mathematical application problems in primary schools, there are two main methods to solve problems: analytical method and comprehensive method, and the thinking directions of these two methods are opposite. One is to solve problems from conditional thinking, and the other is to think about the conditions needed for solving problems from problems. Therefore, teachers can use analysis and synthesis alternately in teaching to cultivate students' reverse thinking. For example, in the teaching process of "multiplication table", teachers can design more "ab+cb" topics to guide students to solve the problems of multiplication table through reverse thinking and deepen their understanding and application.
Reforming exercise design and cultivating students' reverse thinking
The purpose of practice is to digest, absorb and consolidate the knowledge learned in time. Therefore, in order to cultivate students' reverse thinking and improve their problem-solving ability, teachers should design exercises according to students' cognitive level and textbook objectives. For example, after teaching "area calculation of combined graphics", teachers can arrange such a training question according to students' learning situation: "There is a shadow triangle inside a square ABCD, whose vertices are the midpoint E and F of the square AD and CD respectively, and the side length of the square is 1. What is the area of the shadow triangle BEF? "
Under normal circumstances, most students will directly calculate the area of a triangle according to the method of positive thinking, but after some hard thinking, they will find that they can't calculate because they can't calculate the base and height of a triangle. At this time, teachers should guide students to think and analyze problems from different angles. First calculate the graphic area of the blank part, and let the students find that there are three triangles in the blank part. Because these three triangles are all right-angled triangles, the area is easy to calculate, so it is found that the areas of the two slightly larger triangles are equal. Because the area of the square is 1, the areas of the two large triangles are both 0, and the area of the small triangle is also 0, so the area of the shadow triangle is also 0.