I. Understanding the Effectiveness of Mathematics Classroom Teaching The effectiveness of classroom teaching means that teachers can enable students to achieve preset learning results and learn to learn through teaching activities, and at the same time make teachers' own quality develop positively. The concrete manifestations are as follows: in cognition, urge students never to understand and never attend meetings; In terms of ability, gradually improve students' thinking ability, innovation ability and problem-solving ability; Emotionally, it urges students to never like mathematics, to like mathematics and then to love mathematics. Through effective classroom learning, students can learn knowledge and skills that are beneficial to their own development and acquire values and learning methods that will affect their future development. For teachers, through effective classroom teaching, they can feel their own teaching charm and value, enjoy many wonderful moments in the classroom, and let teachers pursue endless mathematics teaching. Second, ways to improve the effectiveness of mathematics classroom teaching 1. Create vivid and effective teaching situations to stimulate students' interest in learning. Confucius, a great educator, once said, "The knower is not as good as the doer, and the doer is not as good as the musician", so the highest realm of learning should be music learning. Improving students' interest in learning is the key to improve the effectiveness of mathematics classroom teaching. Creating suitable teaching situations can greatly improve students' interest in learning mathematics. (1), using suspense questions to create situations In view of students' strong curiosity, teachers use unknown mathematical laws, rules, relationships and facts to create novel suspense situations, showing the extraordinary charm of mathematical knowledge and helping to stimulate students' enthusiasm for exploring knowledge. For example, when teaching the content of "algebraic value", set up a situation: the teacher first asks the students, "You just need to think of a number, then multiply the number you think by 3 plus 8, then multiply the result by 2 minus 16, and tell me the final result, and I will say the number you think within 1 second." The students were surprised and wondered how the teacher knew. (2) Using production and life problems to create situational mathematics comes from and serves life. Therefore, mathematics teaching should closely connect with students' real life, abstract the textbook contents into mathematical problems through familiar examples in life, and show them to students in a situational way to inspire their thinking, eliminate their strangeness and mystery about mathematics, and fully mobilize their enthusiasm for learning. For example, when introducing the "one-yuan-one inequality group", the following questions were thrown out: the confusion of "May Day": during the May Day holiday, the kindergarten teacher gave four sticks and asked to make a triangular kite. My son nailed two pieces of wood A and B together. It is known that the length of A is 10cm, the length of B is 3cm, and the other two blocks are 6cm and 14cm respectively. She chose 6cm, too short, 14cm, too long. I don't know what to do. Is there any way to help solve it? As soon as the question was thrown, the students were enthusiastic. (3) Using stories and games to create situations to integrate mathematical knowledge into interesting stories and games, students' enthusiasm can be easily mobilized. Moreover, it can enhance students' understanding of mathematics, enrich mathematics knowledge, enhance the motivation of learning mathematics, and promote mathematics learning by influencing non-cognitive factors. For example, when introducing the Pythagorean theorem, the author used an animated FLASH to show Pythagoras, a famous mathematician in ancient Greece, having dinner at a friend's house. By observing the floor tiles, he found the quantitative relationship between the three sides of a right triangle ... Through this story, the students' emotions were suddenly mobilized. 2. Feel the inquiry process, and improve students' initiative to participate in mathematics activities. Bruno, a famous American psychologist, said: "Learners should not be passive recipients of information, but active participants in the process of knowledge acquisition." "Exploration is the lifeline of mathematics. Without exploration, there will be no development of mathematics. " Therefore, in teaching, we must return the time to the students to the maximum extent. Let students experience, feel and appreciate mathematics in the process of learning. Only in this way, students can personally experience the joy of their own success, stimulate a strong thirst for knowledge and creativity, and improve their initiative to participate in mathematical activities. In the teaching of "lateral area and Total Area of Cone", the author asked students to make a cone model one day in advance, and said in class: "In this class, we learn" lateral area and Total Area of Cone ". How to find the lateral area of a cone? Can you use the knowledge you have learned to explore the lateral area of a cone with your own cone model as a tool? Can you use letters to express the formula for calculating the side area of the cone? " About 2 minutes later, the author saw that most students found a way to cut the side of the cone into a fan, and some students were at a loss. Ask again: "The side of the cone is a surface, how to find the area of the surface?" "Transform a surface into a plane with the idea of transformation." Most students answered in unison. Another small group of students smiled happily and immediately took action to cut the side of the cone. After about 1 minute, a student shouted happily: "Teacher, I know lateral area of the S cone =" "Is there any other expression?" "Teacher, mine is S-cone lateral area =rl" "I think it is S-cone lateral area =∏rl" "I think it is S-cone lateral area =∏l" The students scrambled to answer. After about five minutes, I asked the representatives of various answers to stand up and explain. "Cut along the mother line of the cone. The side view of the cone is fan-shaped. According to the calculation formula of sector area, the lateral area of S cone is obtained. " "Can you explain what N and L stand for?" "N refers to the degree of the central angle of the sector, and L refers to the generatrix of the cone." "My method is the same as his, but I get lateral area =rl of the S cone, where L is the arc length of the sector and R is the radius of the sector." "My method is the same, but the lateral area of the S cone is equal to ∏ RL, where R is the radius of the cone bottom and L is the generatrix of the cone." "I get lateral area =∏l of the S cone, where H is the height of the cone and L is the generatrix of the cone." "Everyone said makes sense. Which one should I choose as the formula? Why? " "Third, to find the lateral area of a cone, we must know the correlation of the cone. Fourth, although we also know the correlation of the cone, it is more complicated than the third, so I think we should adopt the third as the formula." The author smiled and applauded him. Then, applause thundered in the classroom. 3. Infiltrate mathematical ideas to improve students' thinking quality. "Mathematical thought refers to the spatial form and quantitative relationship of the real world reflected in people's consciousness and produced through thinking activities" and "is the essential understanding of mathematical facts and theories". Junior high school students need to understand mathematical thoughts, such as using letters to represent numbers, combining numbers with shapes, holistic thinking, equation thinking, classified thinking, reduction thinking, analogy thinking and functional thinking. Mathematical thought is the soul of mathematics, which is implicit in mathematical knowledge. With the development of intellectual thinking, it can only be gradually understood and accepted by students. Therefore, in teaching, teachers should take knowledge and examples as carriers, organically infiltrate mathematical thoughts into students, and gradually improve students' thinking quality. For example, when learning "one-dimensional linear inequality group", we can use the solutions of equations to analogize and understand the concepts of inequality group and its solution set, and penetrate analogy thought. Make students migrate on the existing knowledge, and learn new knowledge unconsciously through active participation and exploration and communication. It is intuitive and clear to find the solution set of inequality groups by using the number axis, which permeates the idea of combining numbers and shapes. It is a long-term and meticulous work to solve practical problems, infiltrate modeling ideas and cultivate students' awareness of applying mathematics in teaching. It is impossible for students to fully accept and master it in one or two classes or a few examples, and it should be carried out naturally and imperceptibly in combination with the teaching content. 4. Carefully design exercises to improve students' ability to solve problems flexibly. The new curriculum standard clearly stipulates that students should learn to use the knowledge and methods they have learned to solve simple practical problems. Therefore, practice is an important link in students' learning process. Teachers' purposeful and systematic design of classroom exercises and homework can effectively improve students' ability to solve problems flexibly. (1), design comparative exercises for confusing and error-prone knowledge. In teaching, we should not only pay attention to guiding students to make comparative analysis of some confusing and error-prone knowledge, but also design some exercises in a targeted manner so that students can distinguish and master them through practice and discussion. For example, students can easily understand that the value of k in y=a(x+h)2+k is the abscissa of the intersection of the quadratic function image and the Y axis after learning the "vertex analytic formula of quadratic function". In teaching, guide students to make it clear that K in y=a(x+h)2+k and C in y=ax2+bx+c have different meanings, and then carry out intensive exercises. Find the vertex coordinates of the following function and the intersection coordinates of the image and the Y axis. ( 1)y =-2(x+5)2+4(2)y = 2 x2-5x+4。 This comparative exercise enables students to further distinguish the different functions of the two analytical formulas and greatly reduce the error rate. (2) According to students' different levels, design layered exercises, such as learning the square difference formula. After completing the derivation process of the formula and guiding students to observe the numerical characteristics of the formula and the obtained results, the author shows the exercise set for students to complete: (1)(x+y)(x-y)(a+b)(a-b). (Oral answer) (2)(-x+y)(-x-y), (-a+b) (-a+ 1)(-a- 1), (c-d)(-c-d). (Oral answer) (3)(3n+2m)(3n-2m), (-3m+ 1)(-3m- 1), (xy+2)(xy-2), (3ab+1) (3ab- (5)(a+b)(a-b)(a2+b2), (x+1) (x-1) (x2+1), (-2x+3y) (-3y-2x) (4x. (3) Design variant exercises according to students' thinking characteristics. Variant training should grasp the main line of thinking training, appropriately change the problem situation or change the thinking angle, cultivate students' adaptability and guide students to seek solutions to problems from different ways. Example: It is proved that connecting the midpoints of the sides of a quadrilateral in turn leads to a parallelogram. Variant 1: What quadrilateral is obtained by connecting the midpoints of the sides of parallelogram, rhombus, square and isosceles trapezoid in turn? Variant 2: When the diagonal of a general quadrilateral meets what conditions, is the midpoint connecting the sides in turn a rectangle, a diamond or a square? Will it be trapezoidal? Variant 3: Which quadrilateral connects the midpoints of its sides in turn, and the resulting quadrilateral is rectangular, diamond or square? Through variant exercises, students can better grasp the properties and judgments of various quadrangles, further understand their differences and connections, and improve their ability to solve problems flexibly. . 5. Make full use of multimedia-assisted teaching to effectively optimize classroom teaching. Computer multimedia technology integrates characters, graphics, images, sounds, animations and other functions, and is not limited by time and space. The use of multimedia courseware in teaching can arouse students' learning enthusiasm to the maximum extent, and also express some teaching contents that are difficult for students to understand and understand in language, so that students can understand them intuitively and vividly. What quantity should be used to describe various positional relationships between circles in the teaching of positional relationships between circles? To solve this problem, the author made two animations with the help of multimedia software. Animation 1: Two circles with constant radius make relative translation motion, which makes the two centers blink. During the translation of the circle, students can see that the positions of the two circles are constantly changing. Because the two centers of a circle are constantly flashing, it gives students room for suggestion and association, and realizes that the change of the positional relationship between the two circles should be related to the distance between the centers, thus realizing that the distance between the centers determines the positional relationship between the two circles. Animation 2: Make two circles with fixed centers bigger and smaller. In the process of two circles getting bigger and smaller in turn, students also see the position of the two circles changing constantly, thus realizing that the radius of the two circles also determines the position relationship of the two circles. Through the demonstration of these two animations, students can intuitively realize that the center distance and radius of two circles are used to describe the positional relationship between the two circles, thus breaking through the difficulties in teaching. 6. Pay attention to the cultivation of emotion and improve the internal driving force of effective classroom teaching. Emotion is the experience of people's attitude towards objective things and the link between teachers and students. Good teacher-student relationship and relaxed, harmonious and pleasant classroom teaching atmosphere are the powerful internal motivation to improve the effectiveness of classroom teaching. Only when students feel no pressure and happiness will they be willing to study and give full play to their intelligence. Teachers should first respect and tolerate students, help students with learning difficulties find their own reasons, and encourage students to question boldly. Secondly, teachers should affirm and praise students in time so that students can experience the joy of success. In classroom evaluation, we can use sincere words such as "very good", "great", "not bad", "thinking", "wonderful" and "sorry", and we can also use warm gestures to suggest evaluation. When students answer questions brilliantly, they will give him a thumbs-up or applaud. Teachers' words and deeds, manners and even clothes have an impact on students' behavior, so teachers should show their best mental state to students and devote themselves to teaching activities with full passion. Communicate the relationship between teachers and students, establish deep feelings between teachers and students, so as to achieve the effect of "loving their teachers and believing in their ways", guide students to learn mathematics, and thus comprehensively improve the efficiency of classroom teaching. Three, improve the effectiveness of classroom teaching should pay attention to several issues 1. Fully presupposing classroom teaching is a targeted and planned activity, and the teaching structure needs effective presupposition. Teachers must attach importance to presupposition, study teaching materials deeply, and make full presupposition according to students' reality, so that the classroom will be orderly due to presupposition. If teachers don't study textbooks deeply, they may be tired of dealing with accidents in class and even make jokes. Only by studying the textbooks carefully can teachers grasp the students' learning status at any time, be comfortable with the contents dynamically generated by students in the classroom, and capture valuable information immediately. Therefore, the classroom can be dynamically generated only if it is fully preset before class. 2. Highlighting the Subjective Position of Teachers and Students The new curriculum concept emphasizes that teachers should be organizers, guides and collaborators of subject teaching activities, and effective teaching should aim at the progress and development of students. To implement effective teaching under the new curriculum concept, we should highlight the dominant position of teachers, overcome the teacher-centered and knowledge-centered classroom evaluation concepts, closely focus on the three-dimensional teaching objectives, and stimulate students' initiative and enthusiasm in classroom teaching activities, so as to better implement effective teaching. 3. Timely reflection on teaching Teaching reflection is considered as "the core factor of teachers' professional development and self-growth". The new curriculum attaches great importance to teachers' teaching reflection, which can be divided into three stages according to the teaching process: before teaching, during teaching and after teaching. Reflection before teaching can make teaching a conscious practice; Reflected in teaching, teaching can be carried out with high quality and high efficiency; Reflection after teaching can theorize teaching experience. Teaching reflection will encourage teachers to continuously improve their teaching ability. In a word, effective classroom teaching, as an idea, is a kind of value pursuit and teaching practice mode. We need to constantly explore and study in teaching practice, and gradually improve and improve our teaching philosophy and teaching level.