First, the focus of infiltrating mathematical thinking methods in primary school mathematics teaching
1, infiltration of mathematical thinking methods should strengthen the process.
Infiltrating mathematical thinking method is not injected into the teaching of mathematical knowledge from the outside, because mathematical thinking method is an internal thing related to the occurrence, development and problem-solving process of mathematical knowledge. In teaching, we should not directly point out the mathematical thinking methods used by students, but guide students to experience the mathematical thinking methods imperceptibly in the process of mathematical activities, instead of mechanically copying and telling the whole story. For example, students should write several division formulas with quotient 2. Through observation, we can get the relationship among dividend, divisor and quotient, and boldly guess the law of constant quotient: it may be that dividend and divisor are multiplied or divided by the same number (except zero) at the same time, and the quotient remains unchanged; You can also add and subtract the same number at the same time, and the quotient remains the same. What guess is true? Students use incomplete inductive examples and questions to verify their guesses, and finally get "quotient invariance". Therefore, the process of students' obtaining "quotient invariance" is an empirical process of induction, conjecture and verification, and it is by no means an inductive conjecture verification from the outside. Once students realize this idea, they will think about whether there are similar laws in addition, subtraction, multiplication and division, so as to continue the inquiry process outside the classroom.
2. We should pay attention to repetition when infiltrating mathematical thinking methods.
Pupils' understanding and mastery of mathematical thinking methods has a cognitive process of "from concrete to abstract, from perceptual to rational". Only through repeated infiltration and application can their understanding be improved. For example, students' understanding of extreme thoughts needs a long process of repeated understanding. For example, when they recognize numbers for the first time, they can see that natural numbers 0, 1, 2, 3 ... are "endless" and initially experience that natural numbers are "endless". Students give examples to verify multiplication, division and distribution methods, and use ellipsis or letter symbols to indicate them in the case of countless examples; After teaching the calculation formula of trapezoid area, let the top and bottom of trapezoid approach to zero infinitely, and get the calculation formula of triangle area ... Let students experience infinite meaning many times in limited time and space, and finally reach the understanding of extreme thoughts. At the same time, teachers should slow down in specific teaching, so that students can realize the idea of "infinite number and infinite approximation" in full enumeration and constant experience. For example, some people say that they can't finish painting, and some say that such a small circle needs to be finished. So I asked the students to continue drawing, and I saw that they were a little impatient. Then I asked them to observe the picture of "drawing constantly" in the courseware demonstration, so as to be sure that "the circle has countless symmetry axes". Mathematical thinking method is more abstract and generalized than mathematical knowledge. Only by repeated and long-term infiltration in the teaching process can better results be achieved.
3. Infiltrating mathematical thinking methods should pay attention to systematicness.
The infiltration of mathematical thinking methods should go from shallow to deep, and teachers should make a long-term plan for the degree of excavation, understanding and application of mathematical thinking methods. Generally speaking, every mathematical thinking method always shows some progress with the gradual deepening of mathematical knowledge, so infiltration should reflect the hierarchy of gestation, formation and development. For example, when learning "two-digit plus two-digit", students generally calculate "36+ 17" as "(30+ 10)+(6+7), 36+ 10+7. In the derivation of teaching parallelogram area formula, students should be inspired to consciously use the idea of "conversion" and establish new knowledge learning methods. The area of parallelogram can be transformed into a rectangular area by division and translation, thus integrating all the surface disordered wetting points into a whole.
4. Infiltrating mathematical thinking methods should be made explicit in time.
The mathematical thinking method has a process from vagueness to clarity, from non-molding to molding and then to maturity. In teaching, when the thinking method is hidden, when the thinking method is obvious, we should judge the situation and guide the situation. Generally speaking, in the new teaching of junior and middle schools, exploring knowledge and solving problems are bright lines, while mathematical thinking methods are dark lines. However, in the application of knowledge, class summary or stage review, mathematical thinking methods should be summarized as needed. Senior pupils have learned some basic thinking methods and can call them by their first names. For example, when learning "the division of divisor is decimal", let students try to calculate "6.75÷5.4" first, and many students can't think of a way at the moment. At this time, I suggest: can the divisor be counted if it is an integer? It suddenly dawned on the students that we can use "quotient invariance" to convert the division of divisor into the division of divisor into integer, so I immediately wrote "transformation" on the blackboard to let the students know that the problem to be solved can be attributed to the problem that has been solved by using the idea of "transformation".
Practice shows that the above strategies are a closely related organic whole, and they influence and promote each other. In teaching, we should seize the opportunity, excavate and refine in time, urge students to experience and use thinking methods, and establish a good cognitive structure and a perfect ability structure.
Second, the infiltration of mathematical thinking methods in primary school mathematics teaching
1, reasonably determined in the teaching preset.
Infiltrating mathematical thinking methods, teachers should grasp the effective combination of mathematical knowledge and thinking methods when making teaching presupposition, and embody each mathematical thinking method infiltrated with mathematical knowledge in teaching objectives.
For example, in concept teaching, the introduction of concepts can penetrate multi-case comparison method, the formation of concepts can penetrate abstract generalization method and the penetration of concepts can penetrate classification method. In problem-solving teaching, by revealing the relationship between conditions and problems, common ideas such as reduction, mathematical model and combination of numbers and shapes can be infiltrated.
Sometimes a certain mathematical knowledge contains many ways of thinking, and teachers can focus on it according to the needs and students' cognitive characteristics and determine it reasonably. For example, the new textbook in Shanghai integrates the "operation law and nature", highlights the thinking method of "inductive analogy and mathematical structure", develops students' intuitive thinking, promotes students' learning transfer, and realizes a complete understanding of "operation law and nature". Of course, "running law and nature" should be used in the learning process. Verification and other methods. Only when the main mathematical thinking methods to be infiltrated are determined in the teaching presupposition will teachers study and implement the corresponding teaching strategies and how to infiltrate them? To what extent does it penetrate? Incorporate the infiltration of mathematical thinking methods into teaching objectives (processes and methods), and integrate the requirements of mathematical thinking methods into every link of lesson preparation to reduce blindness and randomness in teaching.
2. Fully experience the formation of knowledge.
Mathematical thinking method is contained in mathematical knowledge, especially in the formation of mathematical knowledge. When learning every mathematical knowledge, try to extract the mathematical thinking method contained in it, that is, let students fully experience it in the process of forming mathematical knowledge.
For example, when I teach the knowledge of "angle", let the students observe "a huge laser emits two laser beams" in the medium, and then the students decide to draw an angle with two beams of light to perceive the definition of "quietness" of the angle and the concept that the size of the angle has nothing to do with the length of the drawn side. Then I asked the students to "make an angle" with tools such as "two pieces of paper and thumbtacks". Inadvertently, the students found that the angle could be rotated. And you can change the angle at will with two paper forks. In this way, the angle is defined as "a ray rotates around its endpoint", which is the definition of "mobility" of the angle and embodies the mathematical thought of motion change. Students have experienced the emergence, formation and development of angles in the activity of "drawing corners", and the mathematical ideas they have learned from it are substantial and profound.
The mathematical thinking method presents a hidden form. In the process of knowledge formation, students experience the methods and hidden ideas carried by knowledge through observation, experiment, abstraction and generalization, so the knowledge they have is vivid and transferable, and their mathematical quality can be qualitatively improved.
3. Strengthen the research of method thinking.
There must be some methods to deal with mathematical content, but mathematical methods are restricted by mathematical ideas. Without the guidance of mathematical thought, mathematical methods are water without sources and trees without roots. Therefore, in the process of thinking about mathematical methods, we should thoroughly study the basic ideas of mathematics.
For example, when I was teaching "See Who's Smart" in grade four, students mainly used the following methods to calculate "1 100÷25": ① vertical calculation 21100 ÷ 25 = (1/. Although methods ②-⑥ have their own advantages, methods ③, ④ and ⑥ use the division of numbers, method ② belongs to the equivalent transformation, and method ⑥ is similar to the "compensation" strategy in estimation, but they are all transformed into easy-to-calculate problems by grasping the characteristics of data and using the learned operation rules and properties, and the same purpose is achieved.
The teaching idea of "algorithm diversification" advocated by the new curriculum is to let students solve problems flexibly by summarizing and optimizing algorithms in the process of learning algorithm diversification, and finally internalize mathematical thinking methods into students' mathematical literacy.
4. Dig carefully in solving problems.
In mathematics teaching, solving problems is the most basic form of activity. Any problem, from putting forward to solving, needs specific mathematical knowledge, but more depends on mathematical thinking methods. Therefore, in the process of exploring and discovering mathematical problems, we should seriously explore mathematical thinking methods.
For example, when I was teaching "Tree Planting" in the third grade, I first asked: On one side of a road with a length of 100 meters, if you plant a tree every 2 meters, how many trees can you plant? Faced with this challenging problem, students speculated one after another, some said to plant 50 trees, and some said to plant 5 1 tree. How many trees are there? Can we start with "planting two or three trees ..." and find out the rules first? With the questions thrown out, the students were lost in thought. If you imagine a hand with five fingers apart as five trees, there is a "gap" (blackboard writing) between every two trees. A * * *, how many gaps are there? The students answered thoughtfully, there are four. If six or seven trees are planted, what is the relationship between the number of trees and the number of intervals? So I inspired students to find the quantitative relationship between the number of trees and the number of intervals when planting at both ends (number of trees = number of intervals+1), which successfully solved the above problems. Then I changed the question to "plant at one end, not at both ends, only plant a few trees", and the students looked for the answer with interest in the same way. Communicate the above problem-solving process to students. We might as well go back to simple problems, then find the rules from the study of simple problems and finally solve complex problems. Through such problem-solving activities, students can feel the important role of thinking methods in solving problems by infiltrating the thinking methods of exploration, induction and mathematical modeling.
Therefore, teachers should consider the design of mathematical problems from the perspective of mathematical thinking methods, try to arrange some problems that will help students to deepen their experience of mathematical thinking methods, and pay attention to guiding students to communicate after solving problems in order to deepen their understanding of problem-solving methods.
5. Refine in time in review and application.
With the deepening of students' understanding of mathematical knowledge, mathematical thinking methods show some progress. In class summary, unit review and knowledge application, teachers should guide students to consciously check their thinking activities, reflect on how they found and solved problems, and what basic thinking methods they used, so as to summarize and refine some mathematical thinking methods in time, so that students can grasp the essence of knowledge from the height of mathematical thinking methods and enhance the value of classroom teaching.
For example, when I was teaching "Review of the Area of Plane Graphics" in Grade Five, I asked students to write out the formulas for calculating the area of various plane graphics (rectangle, square, parallelogram, triangle, trapezoid and diamond), and then asked: How are these formulas derived? Each student chooses 1 ~ 2 kinds of graphs, demonstrates the deduction process with learning tools, and then communicates in groups. After the exchange, I pointed out: Can you organize this knowledge into a knowledge network? When students form a knowledge network, guide them to unify these formulas for calculating the plane graphic area into a trapezoidal formula. Through the above activities, we have deepened our understanding of the idea of "transformation", reorganized students' existing cognitive structure and expanded their mathematical thinking. As the core of the formation of mathematical cognitive structure, mathematical thinking plays an important organizational role.
At the same time, in teaching, if we are only satisfied with the perception and experience of mathematical thinking, it is not enough to confirm that students have understood the mathematical thinking methods used. Only when students apply a certain thinking method to new situations, solve other related problems and be creative can they be sure that students have a deeper understanding of this mathematical method. If students have a preliminary understanding of multiplication, I ask them to rewrite "6+6+6+3" into a simple formula. Most students rewritten "3× 6+3" and "4× 6-3", but a few students wrote the formula of "3×7". Its ingenious operation and unique thinking are valuable to a second-grade child. Teachers should seize the opportunity to induce them to solve problems creatively. For example, after students master the volume calculation of cuboids and cubes, I present an irregular plasticine for students to try different schemes to calculate the volume. After independent thinking and cooperative communication, students found three solutions: ① Pinch a cuboid or cube first, then calculate ② Immerse in the cuboid tank, calculate the volume of water in the rising part, and weigh the plasticine. Divided by the weight (specific gravity) of plasticine per cubic centimeter. The solution comes from students actively applying the idea of "transformation" and then refining it further, so that mathematical thinking methods can be generated together in the process of knowledge and ability formation.
It is not difficult to see from the above practice that if teachers' teaching presupposition is regarded as the early grasp of teaching infiltration, then the formation process of mathematical knowledge, the thinking process of mathematical methods, the discovery process of problem solving and the induction process of review application are the sources for students to form mathematical thinking methods. Students should experience, study, dig and refine themselves, try to figure out and feel mathematical thinking methods, form their own mathematical thinking methods and improve their ability to analyze and solve problems.
Three. Problems and thinking
Bruner, an American educational psychologist, pointed out that mastering basic mathematical thinking methods can make mathematics easier to understand and remember, and mastering basic mathematical thinking methods is a "bright road" to the road to migration. In primary school mathematics teaching, teachers should stand at the height of mathematical thinking method, take mathematical knowledge as the carrier, take into account the age characteristics of primary school students, seize the opportunity, infiltrate mathematical thinking method in time, guide students to actively use the consciousness of mathematical thinking method, and promote the balanced development of students' learning mathematical knowledge and mastering thinking method.
But in the teaching practice research, I am faced with the following problems and thoughts:
1. The new curriculum brings mathematical thinking methods into the teaching target category of "knowledge and skills", which enriches the connotation of mathematical knowledge. However, in the Content and Requirements of primary schools, the teaching requirements of infiltrating mathematical thinking methods are a little general, and they are not clearly refined into specific infiltration contents and requirements suitable for different classes of students, and form a series, which brings certain difficulties to teachers' teaching grasp.
2. At present, the evaluation of primary school students' mathematical learning is still based on the traditional "double basics", and there are few mathematical problems that embody and apply mathematical thinking methods, which is not conducive to investigating the teaching effect of teachers infiltrating mathematical thinking methods and students' mathematical literacy. The evaluation of students' innovative consciousness of using mathematical thinking methods to promote mathematical thinking activities needs further exploration.
3. Primary school mathematics knowledge is relatively simple, but it contains rich mathematical thinking methods. How to deal with the relationship between mathematical knowledge teaching and infiltration of thinking methods, and even form a teaching model suitable for students in different classes to infiltrate mathematical thinking methods, should be deeply thought and practiced.
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