Piaget's constructivism theory holds that students should construct new knowledge on the basis of existing knowledge and experience. The abstraction of mathematical concepts needs to be based on students' existing cognition and pay attention to students' learning process, so teachers should be good at guiding students to gradually abstract and summarize the formal definition of mathematics from their original experience and understanding. For example, there are many ways to reveal the concept of "Time" in the teaching of "Time", but the closer the new knowledge is to the nearest development area of students' cognition, the easier it is for students to understand. Therefore, in this lesson, teachers can use assimilation to guide students to acquire the concept of "time", that is, use students' understanding of "several times" in the existing cognitive structure to assimilate "time". Teachers should encourage students to observe with their own eyes, express in their own language, and interpret the relativity of "times" with their own thinking, so that students can truly understand the relationship between each copy, the number of copies and "times", and let students better establish the concept of "times" and have a deeper understanding.
In addition, in the process of guiding students to understand and master mathematical concepts, teachers can also show the formation process of concepts with rich mathematical history materials, so that students can experience the hard exploration process of mathematicians on mathematical knowledge and mathematical principles. For example, the long process of the formation of the concept of natural numbers, the creation of natural numbers and expressions by different nationalities, and Zu Chongzhi's exploration of pi.
Second, guide students to deeply understand the essence of concepts in mathematical activities.
The so-called understanding of mathematical concept refers to understanding why we should learn this concept, what is the realistic prototype of this concept, and what is the unique mathematical connotation and symbol of this concept, all of which require teachers to guide students to understand step by step. For example, when teaching the concept of natural numbers to first-year students, it is necessary to go through the activities of "counting", while some teachers think that students already have the experience of "counting" in kindergarten, and ignore the teaching of "counting". In fact, preschool children's "counting" still mostly stays at the level of reciting ballads, lacking a deep understanding of logarithm. Without the process of "counting", students' understanding of logarithm is not profound. Therefore, teachers should first design mathematical activities of "number", fully tap the educational value of "number" and let students count in various forms. For example, let students know the number of a set by one number; Enrich students' understanding of logarithm through two or five numbers; Through the changing law of series, students can further understand the characteristics of numbers and discover the inherent law of natural series.
The most basic concept of mathematics is essential and general, and it is the navigator for students to learn mathematics knowledge, while the step-by-step guidance is the golden key to open students' thinking activities. For example, the teaching clip of "Understanding of 10" taught by Wu Zhengxian.
How to grasp the essence of mathematics in teaching Part 2 (1) highlights the realistic background and provides a fulcrum for constructing the algorithm independently.
Students' choice of calculation methods depends more on their real ideas and the most natural understanding accumulated in life practice. It was cold, so Aunt Li went to the wholesale market to wholesale clothes. Take a fancy to a coat 56 yuan and a pair of trousers 44 yuan. If she wants to approve 8 sets of such clothes, how much will it cost for one * * *? What method can be used to answer? "Faced with this problem, students come up with solutions of 56×8+44×8 and (56+44)×8, and then the teacher organizes students to analyze and compare these two methods. Students not only get the same results from the two algorithms, but also find that when the unit price of coat and trousers can add up to 10 or 100, it is more convenient to combine them before multiplication, thus obtaining an optimized solution. Therefore, in teaching, teachers need to create some situations to help students really move from imitation to understanding.
(2) Pay attention to the sense of meaning and lay the foundation for the independent construction of operation rules.
For example, in the above case, after the students get 56×8+44×8=(56+44)×8, the teacher can ask the students while the iron is hot: "If you don't calculate, can you explain why the two solutions are equal with what you have learned before?" Then use the idea of combining numbers and shapes to guide students to understand the equality of two solutions according to the meaning of multiplication. For example, "the school expands the lawn (as shown on the right) and asks for the expanded lawn area." With the help of graphics, students understand that eight 56 plus eight 44 equals eight 100 (that is, 56+44). In the follow-up exercises, teachers need to present this situation repeatedly and variously, and then guide students to look at the formula and think about its original intention.
(3) Gradually abstract and generalize, and build a model of independent construction and operation law.
For example, on the basis of the above teaching, the teacher arranged three kinds of abstract activities: horizontal comparison abstraction, gradual symbol abstraction and old-new comparison abstraction. Horizontal comparison is more abstract (is it true to change the "8 sets" in the example to "20 sets" to make it an equation? Why) breaks away from the abstraction of specific numbers and guides students to preliminarily summarize the laws of multiplication and distribution; Step-by-step symbol abstraction (change "20 episodes" into "C episodes", can it be listed as an equation? Why? What numbers can C stand for here? Changing "56 yuan" into "A" and "44 yuan" into "B", how the equation changes) is divorced from the abstraction of specific situations, from which students are guided to further understand the characteristics of multiplication and division, and the letter expression of multiplication and division is obtained; The contrast between the old and the new is abstract ("a+b" here refers to the price of a suit, what quantity can it represent? Communicate the relationship between old knowledge "the sum of speed" and "the sum of length and width" and new knowledge, break away from the abstraction of specific numbers and specific situations, guide students to improve the cognitive structure of operation rules in communication, and further strengthen their understanding of the characteristics of multiplication and division. The construction of multiplication and distribution law model is naturally generated in the above three abstract processes.
How to grasp the essence of mathematics, implement the third teaching item and carry out effective mathematics activities
Mathematics teaching is an activity teaching, which allows students to experience the process of mathematization and to construct their own mathematical knowledge. The effectiveness of teaching activities depends on whether the activities can arouse students' mathematical thinking and improve their thinking ability. How to make students gain knowledge through hands-on operation in class, how to make operation promote the development of mathematical thinking, and how to make mathematical activities play the greatest mathematical value.
Mathematician Friedenthal said: The knowledge and ability gained through one's own activities are easier to understand and master than those imposed by others. At the same time, they are good at application and can keep a long memory. In the teaching of "Southeast and Northwest", the knowledge of "East-West, North-South Relative" and "Clockwise Arrangement of Southeast and Northwest" is the focus of teaching, which is the method basis for students to find out the other three directions according to one direction. Therefore, in the teaching of this course, two activities are designed to stimulate students' curiosity and exploration spirit, effectively combine external physical activities with internal mathematical thinking, and make students' activities have the taste of mathematics. Students experience these two knowledge points and their application value through activities, which lays a solid foundation for the next activities to find out the other three directions according to one direction. Students participate happily in activities, feel independently, and actively learn and use knowledge. When organizing students to carry out mathematics activities, we should regard students as a living body with rich inner world, independent personal dignity and great potential in life, and design multi-channel, challenging and meaningful mathematics activities, so that students can fully participate in them and help them experience, understand and develop in them.
Give full play to the value-oriented role of teachers.
Because students' knowledge level is in the development stage and their life experience is not rich, their development often cannot be completed spontaneously, which determines that teachers are the soul of the classroom. The realization of any teaching goal is inseparable from students and teachers. Although the classroom is dynamically generated, the teaching process must obey the teacher's preset value pursuit before class (not excluding the conscious adjustment and improvement in the pursuit process) and serve the diversified development of all students. Without the guidance of teachers' value, there can be no high-quality teaching, and students' independent inquiry and cooperation and exchange may lose their direction and become a wandering activity. For example, when a teacher was teaching "oral subtraction within 100 (24- 19)", students reported seven or eight methods after independent thinking. In the process of communication, the teacher repeatedly used "really!" While writing on the blackboard, "Do you have any different opinions?" Guide it. During the whole communication process, the students participated very much and the teachers were very satisfied.
Finally, the teacher said, "Kid, you have so many ways!" " In the future, I will do oral calculations in my favorite way. "In fact, the vast majority of students only understand one of the methods, and almost stay at the original cognitive level, thinking has not been developed accordingly, so the goal of letting students understand and master a variety of oral calculation methods has become empty talk. In this teaching process, teachers are not aware of the internal relations between various methods and the relatively reasonable and concise differences between them, and they are not aware of their responsibility to guide students to make comparative classification, and then make choices and self-adjustments on this basis, so that students' construction activities become meaningful rather than chaotic. Formal opening and liberalization can only bring superficial excitement and lack of effectiveness. This requires teachers to accurately grasp students' learning dynamics, so as to "do it when it is time", that is, to intervene in time and give full play to teachers' value leading role.