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 methods is not injected into the teaching of mathematical knowledge from the outside. Because mathematical thinking method is an intrinsic thing related to the occurrence and development of mathematical knowledge and the process of solving problems. In teaching, we should not directly point out the applied mathematical thinking methods, but guide students to experience the mathematical thinking methods in the process of mathematical activities in a subtle way to avoid copying them mechanically. For example, students write several division formulas with quotient 2. Through observation, we can get the relationship between 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 also 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 wonder whether there are similar laws in addition, subtraction, multiplication and division, thus extending the inquiry process to extracurricular activities.
2. We should pay attention to repetition when infiltrating mathematical thinking methods.
Primary school students' understanding and mastery of mathematical thinking methods has a cognitive process of "from concrete to abstract, from perceptual to rational", and their understanding can be enhanced through repeated infiltration and application. For example, students' understanding of extreme thoughts requires a long process of repeated understanding. For example, when only recognizing numbers, let students see that the natural numbers 0, 1, 2, 3 ... are "endless" and initially experience "infinite"; 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 trapezoid area calculation formula, let the upper and lower sides of the trapezoid approach to zero infinitely, and get the triangle area calculation formula ... Let the students experience the meaning of "infinity" many times in a limited time and space, and finally reach the understanding of the limit thought. At the same time, in specific teaching, teachers should slow down and let students realize the idea of "infinite number and infinite approximation" in full enumeration and continuous experience. For example, when teaching "Understanding Circle", after students drew several symmetry axes, I asked them if they could finish drawing such symmetry axes. Some people say that you can't finish painting, others say that you should finish painting such a small circle. So I asked the students to continue painting, and I saw that the students were a little impatient. Then I asked them to observe the picture of "painting without stopping" 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 a certain degree of progress with the gradual deepening of mathematical knowledge, so it should reflect the hierarchy of gestation, formation and development when infiltrating. For example, when organizing the study of "two-digit plus two-digit", we should reflect the ideological gestation period of "returning": students generally have "(30+ 17)+(6+7), 36+/kloc-0+7,36+. In the teaching of two-digit multiplication and division, students should be gradually guided to have a clear understanding of this; In the derivation teaching of parallelogram area formula, students should be inspired to consciously use the idea of "transformation" to establish a method of learning new knowledge. The area of parallelogram can be converted into the area of rectangle by division and translation. In this way, the various infiltration points on the surface are integrated 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 unformed to shaped and then to mature. In teaching, we should assess the situation and improvise when the way of thinking is hidden and revealed. Generally speaking, in the new teaching of middle and lower grades, exploring knowledge and solving problems is a bright line, and the mathematical thinking method is a dark line. However, in the application of knowledge, class summary or stage review, mathematical thinking methods should be summarized and summarized as needed. Senior pupils have learned some basic thinking methods and can call them by their first names. For example, when learning "divisor is the division of decimals", let students try to calculate "6.75÷5.4" first, but many students can't figure out a way at the moment. At this time, I prompted: Can the divisor be an integer? It suddenly dawned on the students that "quotient invariance" can be used to convert "division of divisor is decimal" into "division of divisor is integer" to solve the problem, 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 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, the common ideas of reduction, mathematical model and combination of numbers and shapes in mathematical problem-solving are infiltrated.
Sometimes a certain mathematical knowledge contains a variety of thinking methods, 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, in the process of learning, we also need to use methods such as "observation, guessing and verification". 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", I ask the students to observe "a huge laser emits two laser beams" in the medium first, and then the students decide to draw an angle with two beams of light, and perceive the definition of "static" angle and the concept that the size of the angle has nothing to do with the length of the drawn side. Then let the students "make corners" with tools such as "two pieces of paper and thumbtacks". Inadvertently, the students found that the corner can rotate and change with the size of two pieces of paper. In this way, "angle" is defined as "a ray rotates around its endpoint", which is the definition of "mobility" of angle, which embodies the mathematical thought of movement and change. Students have experienced the emergence, formation and development of "corners" in the activities of "drawing and making corners", and the mathematical ideas they have learned from them are substantial and profound.
The mathematical thinking method presents a hidden form. In the process of knowledge formation, students experience knowledge-loaded methods and implicit ideas through observation, experiment, abstraction and generalization, so the knowledge they master 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/. The results show that method ① is a general method and methods ②-⑥ are ingenious methods. 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 all methods transform data into easy-to-calculate problems by grasping the characteristics of data and using the learned operation rules and properties, thus achieving the same goal. Students' evaluation and reflection on various methods is to deepen the mathematical thought behind the methods, so as to have an essential grasp of mathematical knowledge and methods.
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? Can we start with "planting two or three trees ..." and find out the rules first? With the questions raised, the students were lost in thought. If you look at five trees with your fingers open, there is a "gap" (blackboard writing) between every two trees. A * * *, how many gaps are there? The students answered thoughtfully that there were four. If you plant 6 trees and 7 trees, what is the relationship between the number of trees and the number of intervals? So I inspired students to put a pendulum, draw a picture and discuss it, and found 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 change the question to "plant only one head, plant several trees without planting both heads", and the students look for the answer with interest in the same way. The above-mentioned problem-solving process conveys a strategy to students: when encountering complex problems, they might as well retreat to simplicity, then find the law 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 deepen the 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, and 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 derivation 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 area of plane graphics into trapezoidal formulas for calculating the area. Through the above activities, we have deepened our understanding of the idea of "conversion", reorganized students' existing cognitive structure and expanded their mathematical thinking. As the core of the formation of mathematical cognitive structure, mathematical thinking method has played 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 affirm that students have understood the mathematical thinking methods used. Only when students apply a certain thinking method to a new situation can they solve other related problems and be creative, and can they be sure that they have a deeper understanding of this mathematical method. If students have a preliminary understanding of multiplication, I will ask them to rewrite "6+6+6+3" into a simple formula. Most students have rewritten "3× 6+3" and "4× 6-3", but some students have written the formula of "3×7". Its ingenious operation and unique thinking are very valuable to a second-grade child. Secondly, when students' creativity is in a good state of preparation, 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, the students found three solutions: ① Knead into a cuboid or cube, then calculate the volume of water immersed in the cuboid water tank, and calculate the weight of plasticine, and then divide 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. In the process of learning, students should experience, explore, refine, ponder 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 understanding basic mathematical thinking and methods is a "bright road" to the road of migration. In primary school mathematics teaching, teachers should stand at the height of mathematical thinking method, take mathematical knowledge as the carrier, consider the age characteristics of primary school students, seize the opportunity, infiltrate mathematical thinking method in time, guide students to actively use mathematical thinking method, promote the balanced development of students' learning mathematical knowledge and mastering thinking method, and lay a solid foundation for them to learn mathematics well in the future.
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 examining 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 of different classes to infiltrate mathematical thinking methods, should be deeply thought and practiced.
Please adopt
If you agree with my answer, please adopt it in time.
~ If you agree with my answer, please click the button "Adopt as Satisfied Answer" in time.
~ ~ friends who ask questions on mobile phones can only be satisfied with the evaluation points in the upper right corner of the client.
~ Your adoption is my motivation.
~~O(∩_∩)O, remember to praise the adoption and help each other.