The concept has its inevitability. It is necessary to grasp the background of the emergence of concepts, let students understand the reasons for the emergence, development and evolution of mathematical concepts and the internal relations between mathematical concepts hidden in these reasons, and reflect the role of mathematical concepts in the overall coherence of mathematical thinking.
Therefore, when teaching new concepts, teachers can analyze the background of concepts and find interesting and vivid entry points suitable for students' understanding, so that students can understand new concepts more easily and discover new knowledge more easily, so that students can have more opportunities to participate in discovering and establishing new concepts, join this creative activity and feel the beauty of harmony, coherence, rigor and usefulness in mathematics. The following are some methods used in concept teaching.
First, starting from the background of the concept, it is getting deeper and deeper.
The concept of logarithm is a very abstract concept that students encounter in mathematics learning. The direct teaching method will make it difficult for students to understand. In fact, if we analyze the background of logarithm, we can find that it is a new operation that will inevitably appear after the mathematical operation develops to a certain stage. When addition develops to a certain extent, subtraction will inevitably be introduced, and when power develops to a certain stage, squares will inevitably appear. Logarithm is also inevitable for the calculation demand in production and life. If the background and operation methods of these concepts are tabulated, new concepts will naturally be formed in the process of comparison, which will be easy for students to accept and understand.
Teachers can set up such a teaching lead-in process: first, ask two questions: 1 and 1 cells divide into two cells at a time. 1 How many times does it take for a cell to divide into 128? 2. A person's original annual salary is 10,000. Suppose his salary increases by 65,438+00% every year, how many years later will his annual salary double?
In these two examples, the operation used is to solve the exponential equation: 1, 0, 2,. But the answer to the first question is a special value, and no new operation is needed. The answer to the second question is not a special value. In the existing operation, it is impossible to work out the answer. How to solve this problem?
Then, the teacher put forward several reciprocal operations for comparison, such as: 3+x= 10 x= 10-3, 5=8 x=,.
In the following teaching, we can naturally transform exponential expression into logarithmic expression x=, and introduce a new concept of operation, pointing out that the relationship between exponential expression and logarithmic expression (1) is equivalent. (2) They just write and read differently. A, B and N have different names and different positions, but they represent the same number, with the same meaning and the same range of numbers. As long as you firmly remember the way and position of letters A, B and N in exponential and logarithmic expressions, you can freely exchange exponential and logarithmic expressions. In this process, the relationship between exponential logarithm and square root of addition, subtraction, multiplication and division is similar, and the comparison between these concepts should run through the teaching and be easy for students to understand.
Second, starting from the concept of life background, create a learning situation.
Many mathematical concepts are the products of people's highly abstract generalization of things in long-term real life. They are based on concrete materials and have vivid prototypes. Teachers should be good at creating good learning situations, stimulating students' interest in learning through various ways, and making students feel that these abstract mathematical concepts are around and within reach.
The concept of geometric series comes directly from the concept of life. In the process of teaching, there are all kinds of real-life examples, such as common cell division problems, store discounts, the weight of radioactive materials, bank interest rates, and choosing a suitable repayment method for one's family, which can be easily interspersed in the process of explaining and consolidating concepts.
In order to give full play to the enthusiasm of students, I also designed an interesting problem situation to introduce the concept of geometric series:
Achilles (a hero who is good at running in Greek mythology) races with the tortoise. The tortoise leads by 65,438+0 miles. Achilles' speed is 10 times that of tortoise. When he caught up with 1 mile, the turtle rushed forward. When he chased the forest, the tortoise went forward. When he caught up with the forest, the tortoise went on. ...
(1) Write the distance that Achilles and the tortoise walked in the same time period;
(2) Can Achilles catch up with the tortoise?
Let the students observe the characteristics of these two series and lead to the definition of geometric series. The students are very interested, enthusiastic and active, and the classroom atmosphere is very active.
Third, from the historical background of the concept, stimulate interest.
The concepts of complex number and imaginary number have a long historical background and are the inevitable products of the development of numbers to a certain stage. For a long time, people can't find the quantity represented by imaginary number and complex number in real life, and the life prototype of imaginary number can't be found in students' limited knowledge structure, so it is difficult for students to fully understand it. Therefore, when explaining these two concepts, we can simply explain the development history of numbers and the appearance of imaginary numbers and complex numbers in order to make it clear.
Since primitive people distributed food, natural numbers first appeared, and then scores appeared. After a long development of numbers, people have found many numbers that cannot be written by the ratio of two integers, such as pi. People write them as π and call them irrational numbers. In the19th century, it is often necessary to open squares in operation. For example, if the number opened is negative, will there be a solution to this problem? If there is no solution, mathematical operation is like walking into a dead end. In this way, students can be involved in teaching, think with the end of the story, and then introduce a new concept: mathematicians stipulate that the symbol "I" is used to represent the square root of "-1", that is, =- 1, and the imaginary number is born. Real numbers and imaginary numbers are combined to write A+.
Fourth, starting from the concept of professional background, stress practicality.
Many mathematical concepts are also widely used in other professional fields. Combining mathematics knowledge with other professional knowledge can make students fully realize the importance of mathematics learning.
The concept of trigonometric function has important applications in many professional fields. In physics, simple harmonious motion, orbital motion of stars, peak and valley electricity; Psychologically and physiologically, the periodic fluctuation of emotions, the periodic changes of intelligence and physical strength, and the blood pressure of a day; In astronomical geography, the law of temperature change, the law of moon deficiency and full moon, and the law of tide rise and fall; In daily life, the change of wheels is inseparable from trigonometric functions.
Therefore, in the application course of trigonometric function, we can design some practical problems with periodic changes, let students build simple trigonometric function models, cultivate students' abilities of mathematical modeling, problem analysis, combination of numbers and shapes, abstract generalization, experience the value and role of mathematics in solving practical problems, and cultivate students' spirit of diligent thinking and courage to explore.
Students can only learn new concepts on the basis of existing knowledge, so teachers must pay attention to the knowledge structure of textbook design in teaching. They should be bold and innovative, and design each concept lesson according to the actual situation of students.