1 How to improve mathematical understanding ability
Using comparison skillfully and internalizing the essential connotation of knowledge
Learning should enable students to gradually clarify the essential connotation of knowledge in the process of experiencing the formation of knowledge. Famous educators Bowden and Ma once pointed out that when one aspect of a phenomenon or thing changes while other aspects remain unchanged, those changed aspects are easily recognized by people. Therefore, in the teaching process, according to the characteristics of some teaching contents, we should properly and skillfully use comparative strategies, so that students can internalize the essential connotation of knowledge independently in the process of changing and unchanging, so as to make students' mathematical thinking more meticulous, help students grasp what they have learned more accurately and improve their mathematical understanding ability. For example, when I was teaching the relationship between triangles, the second volume of the fourth grade of Jiangsu Education Press, I started teaching by using the mode of group cooperation and inquiry. In the process of independent inquiry, I provided each study group with sticks of different specifications, such as 15cm, 8cm, 7cm, 5cm, 4cm, 3cm, etc., so that students can randomly select three sticks to surround each other through independent operation, observation and research to see what can form a triangle. What cannot be surrounded? And fill in your own research results in the activity list of "Relationship among Three Sides of Triangle" (see table 1).
In the communication session, I mainly guide students to analyze: What is the relationship between the three sticks that can form a triangle? What are the characteristics of three sticks that cannot form a triangle? Because of the students' operation and comparative experience just now, I quickly found out that "the sum of the lengths of two short bars is greater than the third bar, and a triangle can be formed, otherwise it can't". Then I organize students to think: If you choose an 8 cm long stick and two sticks of other specifications, how many different choices do you have? The main purpose of this operation activity is to let students further clarify the relationship between the three sides of a triangle on the basis of "unchanged" and "changed". Such calculation and comparison not only effectively realize the internalization of knowledge, but also effectively improve students' mathematical thinking ability such as inductive reasoning.
Skillfully using comparison method to improve the efficiency of internship training
Practice is an effective means to help students consolidate what they have learned and improve their knowledge and skills. However, the current situation of students' homework is really worrying: first, some teachers lack the necessary careful selection of homework, which leads to a serious shortage of "gold content" in exercise questions; Secondly, some teachers take "practice makes perfect" as a magic weapon, leading their students to climb up and down in the sea of questions, exhausted physically and mentally. However, their "painstaking efforts" often result in "endless suffering" instead of "everyone is happy". Therefore, I think it is particularly necessary to establish a new concept of mathematics homework, skillfully use comparative ideas, design problem sets and improve the quality of students' mathematics homework.
For example, after studying the unit "The Meaning of Fractions" in the second volume of the fifth grade of Jiangsu Education Publishing House, I found that students were interested in "cutting a 3-meter-long rope into 6 sections on average, how many meters is each section?" Failure to master a class of topics is often related to the question "how long is each piece of this rope?" A kind of practical problem is confusing. So, in the unit arrangement and review, I designed a set of exercises for students: ① Cut a 40-meter-long rope into 8 sections on average, how long is each section? ② Cut a 4-meter-long rope into 8 sections on average. How many meters is each section? ③ Cut a 4-meter-long rope into 8 sections on average. How long is each part? In this set of exercises, students can easily find that the number of the first question and the second question is the same, and both require the specific length of each paragraph. The main difference is that the total length of the rope causes the length change of each section, so the thinking method of these two problems is the same, that is, the total length of the rope divided by the average number of cut sections. The known conditions of the second and third sub-topics are the same, but the difference is that the second sub-topic asks about the specific length of each paragraph, while the third sub-topic asks about the relationship between each paragraph and the total length (the average of the total paragraphs). With this contrast, it is naturally more effective to lead students to finish their homework. In the process of "hyperlink" exercises, students not only master the problem-solving strategies of these exercises, but also gradually improve the "knowledge network" and help them develop good reflective learning habits. Space can make the classroom full of active learning atmosphere, and also help students acquire useful knowledge through channels other than textbooks, thus improving the efficiency of mathematics teaching.
2 Mathematics teaching methods
Set three-dimensional goals reasonably, and make clear the key points and difficulties.
The three-dimensional teaching objectives put forward by the mathematics curriculum standard of ordinary senior high schools are: knowledge and skills, process and method, emotional attitude and values. Knowledge and skill goals include basic knowledge that students need to know, understand and understand, basic principle goals and basic skill goals that students must achieve; The goals of process and method include realizing the inquiry process and method in mathematical science, optimizing students' learning process, and emphasizing students' experience of exploring and acquiring new knowledge;
The goals of emotional attitude and values include students' interest and enthusiasm in learning, the spirit of overcoming difficulties, understanding the beauty of mathematics and shaping students' personality. However, these teaching objectives are greedy, comprehensive, far-fetched and difficult to implement. The reason for this situation is that teachers do not attach importance to the formulation of classroom teaching objectives, and think that this is a "retreat" work, confuse curriculum objectives with classroom teaching objectives, and "copy objectives" from reference materials. Therefore, if teachers want to formulate three-dimensional goals reasonably, they should make clear the hierarchy of goals: mathematics curriculum goals, module teaching goals, unit teaching goals and classroom teaching goals.
Understand the teaching materials thoroughly and make full use of them.
The new mathematics curriculum in senior high school is written by experts with profound theoretical and practical level of mathematics specialty and education and teaching. They are compiled according to the actual situation of students after systematic analysis and careful consideration of teaching situation. The content of the textbook is well arranged, easy to understand and contains rich information. Teachers need to dig out the information contained in it and sort it out systematically, so as to fully display the connection between knowledge, clarify the ideas and intentions of teaching materials, and fully display the educational contents such as knowledge, skills, emotion and value contained in teaching materials. At the same time, on this basis, teachers should fully tap the teaching materials, be loyal to the teaching materials, not stick to the teaching materials and surpass them. Teachers can combine the actual situation of students in this class to design the most suitable topic for students, so as to inspire and induce students to have a deeper experience and understanding of the teaching materials and let students "walk into the teaching materials and walk out of the teaching materials".
For example, when teaching the image of "function y=Asin ()", teachers can let students read books and study independently from page 34 to page 37 of the textbook, and discuss their learning experiences and problems in groups; Then students report, and teachers and students jointly reveal ideas and methods; After that, self-study the example 1, and students draw the transformation flow chart; Finally, test the feedback and let the students complete the exercise on page 39 of the textbook. The teaching materials of this lesson, from examples to feedback exercises, are all taken from textbooks, which have a good demonstration role and are also classics of thinking training.
3 Cultivation of interest in mathematics learning
Cultivate interest in learning with vivid language
The teaching content of mathematics is abstract, boring and tasteless. It has no vivid language and vivid stories, and it is not easy to arouse students' interest in mathematics. So when teaching students to recognize and count, I use concrete images and some interesting stories to stimulate students' interest. For example, when reviewing the addition and subtraction within 10, I raised a question in the form of a story: Little Rabbit's good friend celebrated his birthday, and Little Bear lived in the deep forest and arrived early in the morning. He brought two barrels of the best honey to the rabbit.
Look, after a distance, he shouted, "rabbit, rabbit, open the door quickly." What do you think I brought you? " The rabbit has smelled the sweetness of honey, so she hurried out to meet her. "Thank you, thank you, please come in and sit down!" "In a short time, Xiaohua Mall came also and gave the rabbit five apples. Here comes the little monkey. He picked four peaches from the orchard and gave them to her. The chicken came, too, but she didn't send anything and ate three apples. The rabbit is unhappy. Please count the children. How much fruit has this rabbit collected? How many apples are there at last? After listening to this vivid story, the students have a high enthusiasm for learning. Soon they listed the formula and worked out the result: 5+4=95-3=2. Later, I encouraged students to make up stories and learn arithmetic by making up short stories and using pictures in books. Students study happily and actively.
Cultivate learning interest with praise and evaluation
A sense of honor can enhance interest in learning. Primary school students have a strong sense of honor, and the maintenance of their interest in learning depends largely on the social effects obtained through learning. They are often praised by teachers, parents, brothers and sisters, classmates and friends because of their good grades, which causes a sense of honor. They study harder to keep their honor. Students should be given correct and appropriate praise in teaching. When students answer questions, stare at them and encourage them to speak boldly with expectant eyes. We should pay attention to seize every opportunity to give praise and encouragement. Even a word of praise, a few words of encouragement and a small red flower can better stimulate students' interest in learning. In particular, students who are struggling with their studies should be appropriately reduced, praise should be emphasized and progress should be encouraged. Praise is a concrete manifestation of teachers' love for their jobs and students, and it is also an important means to maintain students' interest and give full play to their potential.
At the same time, I think appropriate comments can not only guide students' learning methods, but also stimulate students' interest in learning and strengthen their learning motivation. For example, for some well-done exercise books, you might as well write "The method is too good, but be careful!" "How clever! You must have a brilliant idea, because you are the pride of the teacher! " Praise talented students, but don't scold poor students. On the contrary, we should seize their bright spots and give them timely encouragement. For example, "Work hard, make progress and keep working hard!" "Seeing that you are making progress, the teacher is really happy for you because you have worked hard." "You can do it, and the teacher believes in you!" Such emotional comments make students feel the teacher's concern for him and full of hope, thus gradually generating strong interest.
4. Students' divergent thinking ability
Cultivating students' divergent thinking ability is inseparable from learning calculation methods and mastering problem-solving methods, and also requires practice. Thinking is closely related to problem-solving process. The most effective way to cultivate thinking ability is through problem-solving practice. Therefore, the quality of exercise design is the key to promote the development of students' thinking ability. In teaching, we must reasonably arrange exercises in and after class according to class differences and students' characteristics, and the training questions must be targeted and effective.
Stimulating students' problem-solving ability is also an effective way to cultivate students' divergent thinking ability. Solving problems is the core of mathematics, and the ability to solve problems is an important symbol of students' mathematical literacy. In teaching, teachers should focus on students' life experience and practical experience, open students' horizons, broaden students' learning space, maximize students' potential, make students experience the close connection between mathematics and daily life, cultivate students' ability to find mathematical problems from surrounding situations, apply what they have learned to solve practical problems, develop students' application consciousness and form problem-solving strategies.
Paulia once said that the best way for students to acquire any knowledge is to discover it by themselves. Because this discovery is the most profound to understand, and it is also the easiest to grasp the internal laws and connections. Therefore, in the process of classroom teaching, teachers should fully guide and, more importantly, give students enough time to think and experience, so that students' learning process can become a dynamic process, and students can actively and openly explore and discover according to their own experiences and their own ways of thinking, so as to master problem-solving strategies, experience the diversity of problem-solving strategies and improve their ability to solve practical problems.
The generality, problem and logic of thinking are important manifestations of students' thinking ability. Therefore, we should grasp every link in teaching, strive to cultivate students' thinking ability, and provide a powerful carrier for problem-solving learning. Students' thinking ability will develop with age, and students' problem-solving ability and mathematical thinking ability are complementary and inseparable. The improvement of problem-solving ability will promote the development of thinking ability, and the development of thinking ability will also make students' problem-solving ability go up a storey still higher. Therefore, we should cultivate students' mathematical thinking ability in the whole teaching. This infiltration should extend from the classroom to the outside, from the lower grades to the higher grades, so as to make it a coherent whole, so as to achieve the purpose of cultivating students' thinking ability and strengthen the guidance of students' thinking strategies.