Before learning each new lesson, I ask students to teach themselves the textbooks. I don't repeat what they have understood and mastered through classroom self-study. For the knowledge points that they think are simple but contain philosophy, or that they didn't pay attention to at all during the preview, we can stimulate students' curiosity and thirst for knowledge by setting "traps" or carefully designing challenging questions, attract students' attention, and make students actively participate in the learning process, explore problems and break through difficulties. For example, when I was teaching the positional relationship between straight lines and circles, as soon as the topic was introduced, some students volunteered to go to the podium to explain. He used compasses and triangles to demonstrate the graphic representation, text representation and two judgment methods of three positional relationships between straight lines and circles on the blackboard. When this classmate returned to his seat, other students and I couldn't help applauding him. Because his paintings are not very beautiful, the vertical section from the center to the straight line and the radius of the circle are marked with pens of different colors, and the whole table is in good order. Just as the students thought that all the knowledge points in this lesson were mastered and they were ready to "sharpen their knives", I asked, "Do you still have any questions about the positional relationship between straight lines and circles? If not (before I finished, the following students all replied in unison: no! ), then let me ask you a question: Why do straight lines and circles only have these three positional relationships? Are there any other positional relationships? "
Answer: No, because only these three positional relationships are drawn in the textbook.
The teacher asked again: why do you dare to conclude that there is no fourth positional relationship after drawing these three positional relationships?
Answer: Straight lines and circles have nothing in common-they are separated, but there is only one thing in common-tangency, and there are two things in common-intersection, but straight lines and circles cannot have three things in common.
The teacher asked: Why?
Because no three points on a circle are straight. I and other students in the class applauded him for his quick wits. )
The teacher also asked: A straight line and a circle cannot have three things in common. Do other students have other methods of argument?
After this question was asked, the whole classroom became "the dawn here is quiet". After about two or three minutes, a student finally stood up and answered loudly: because the most unknown in their equation is quadratic, and the quadratic equation with one variable has at most two unequal real roots. The words sound just fell and thunderous applause rang out in the classroom. As soon as this classmate sat down, another classmate stood up and answered: According to the principle of inequality, there are only three relationships between the size of any two real numbers, so there are only three relationships between the distance from the center of the circle to a straight line and the size of the radius. After listening to the students' wonderful answers, do you still need to emphasize the mathematical thought of this lesson? Will students still find this course boring? Do you need to worry about the effectiveness of this class?
2. Guide students to think positively and stimulate their interest in learning.
The problem is the core of mathematics. Students must learn to face a situation, be good at grasping its essence and ask core questions, which is the most important quality. In classroom teaching, teachers should fully respect students' cognitive rules, instead of always adopting a single mode of "teachers ask students to answer", they should actively advocate the interactive mode of "students ask students to answer" and "students ask teachers to answer". This can not only fully expose the problems among students, but also make students change from passive acceptance of knowledge to active exploration and acquisition of knowledge, so that students can experience the whole process of knowledge generation, development and generation and stimulate their interest in learning. For example, after learning the definition of arithmetic progression, some students asked: Teacher, since there is arithmetic progression, are there equal sum series, equal product series and equal quotient series? Then the teacher and the students discuss this problem together, which not only makes the students firmly grasp the essence of the definition of sequence, but also paves the way for the follow-up study and broadens the students' knowledge. For another example, after learning arithmetic progression's definition and solving the exercise "Know arithmetic progression 3, b, c, -9, and find the values of b and c" with the method of equation, some students asked: If "Know arithmetic progression 3, b, c, d, -9, and find the values of b, c and d", is there any other way besides understanding the equation? When the students directly calculate the tolerance with the first two terms and the last two terms, and then calculate the values of B, C and D one by one according to arithmetic progression's definition, some students immediately said: Don't say to insert two or three numbers between 3 and -9. Even if the numbers are inserted, I can break them one by one as long as they become arithmetic progression. This question not only makes students deeply realize the importance of tolerance D of basic quantity in arithmetic progression, but also cultivates the divergence of students' thinking. Untie the reins that bind students' thinking, give them enough time and space, and let their sparks of wisdom collide with passion. Why don't students fall in love with math class?
3. Presupposition questions and develop students' autonomous learning potential.