What is the basis of the new stage of senior high school mathematics teaching? What kind of foundation should be laid? How to lay a good foundation? These problems are what we educators must make clear when implementing the new curriculum. This paper discusses these aspects.
The first is the correct positioning of "double bases"
According to the concept of the new curriculum, basic knowledge and basic skills should keep pace with the times. So, how to correctly position today's "double base"? The author believes that the definition of "double base" should consider two aspects: foundation and development.
(A) pay attention to the new changes in curriculum objectives
Curriculum standards put forward three requirements for mathematics curriculum objectives. The first level is knowledge education, which emphasizes that students should not only acquire the necessary basic knowledge and skills, but also understand its context and understand the mathematical ideas and methods contained in it. The second level is the cultivation and education of students' mathematical quality and ability. In addition to improving students' mathematical thinking ability (including five basic abilities of spatial imagination, abstract generalization, reasoning and argumentation, operational solution and data processing), it is also proposed to improve students' abilities of asking questions, analyzing and solving problems, mathematical expression and communication, independently acquiring mathematical knowledge, developing students' mathematical application and innovation consciousness, and being able to think about quantitative relations and mathematical models in objective things. The third level is the educational level of cultivating non-intellectual quality. It is proposed to stimulate interest, build confidence, form a scientific attitude of seeking truth from facts and persistent research spirit, form the habit of critical thinking, understand the scientific value and humanistic value of mathematics, and establish a dialectical materialist world outlook. This is very different from before.
(2) Pay attention to the new changes in the definition of knowledge and the formulation of ability.
The definition of mathematics in the curriculum standard is more incisive, pointing out that "mathematics is a science that studies the relationship between spatial form and quantity" Compared with the original statement that "mathematics is a science to study the relationship between spatial form and quantity in the real world", it embodies a new understanding and definition of the object of mathematical research, making surreal form and relationship a part of the object of mathematical research. The basic knowledge of mathematics is no longer limited to the concepts, properties, laws, formulas and theorems in mathematics, and the mathematical thinking method reflected from it is also defined in the basic knowledge, which is implicit knowledge contained in explicit knowledge. As a basic knowledge, it is more important to learn and master its thinking method. In the formulation of ability, new requirements are put forward for ability training on the basis of the original. While paying attention to improving students' mathematical thinking abilities such as spatial imagination, intuitive guessing, inductive analogy, abstract generalization, symbolic representation, operational solution, data processing, deductive proof, reflection and construction, we also emphasize cultivating students' ability to raise, analyze and solve problems with mathematics, to express and communicate with mathematics, to acquire new mathematical knowledge, to explore with mathematics, and to be aware of mathematical application and innovation, hoping to become a mathematical consciousness and consciously understand objective things.
(C) Pay attention to the new changes in teaching content
According to the concept of new curriculum standards, high school mathematics curriculum should be diversified and selective, so that different students can get different development in mathematics. Therefore, great changes have taken place in the division of courses, the determination of content and the adjustment of structure. Mathematics course is divided into compulsory course and elective course, and the compulsory course consists of five modules. The contents of the five modules cover the main parts of traditional basic knowledge and basic skills in senior high school. The difference is that while laying a good foundation, we further emphasize the occurrence, development process and practical application of this knowledge, and there is no excessive demand for skills and difficulty. The structure of some reserved contents has also changed, such as the integration and appropriate simplification of deformation such as analytic geometry, solid geometry and trigonometric constancy: basic contents such as vectors, algorithms and probabilities have been added, and the most basic knowledge of data processing and statistics has been regarded as the new basic knowledge and skills of mathematics, and oral and written mathematical expressions have also been listed as the basic skills of learning mathematics well; Trivial calculations, artificial technical problems, and excessive emphasis on details have all been deleted. The contents of mathematical inquiry, mathematical modeling and mathematical culture are set up, which requires that the ideas of mathematical inquiry and mathematical modeling should be infiltrated into each module and theme content in different forms, the contents of mathematical culture should be organically combined with the contents of each module, and the contents of emotion, attitude and values should be integrated into the curriculum.
(4) Positioning according to changes
The above changes show that with the development of the times and mathematics, the basic knowledge and skills of high school mathematics have changed. The so-called "double basics" should be the organic integration of various elements and the basic literacy necessary for students' lifelong development. A solid foundation means not only the accumulation of knowledge, but also the "double foundation" means not only knowledge and skills. Innovative consciousness, application consciousness, practical ability, ability to think and judge with mathematical methods, life planning ability, scientific spirit, critical thinking habits, entrepreneurial consciousness, etc. It is also the foundation, even more important. There is also a strong interest in learning, a strong thirst for knowledge, a positive spirit of exploration and emotional attitude, the ability to collect and process information, the ability to acquire new knowledge, and the ability to communicate and cooperate. These are the basic connotations and foundations of laying a solid foundation for students in an all-round way. Only by integrating with the learning of knowledge and skills can we promote each other and form a new "double base" that meets the requirements of the times.
Second, the "double basics" thought and several relations
How to lay a good foundation for students in the new stage of mathematics teaching in senior high school? In view of the change of the connotation of "double basics", its methods and ideas should also change accordingly. It is necessary to clarify the thinking of mathematics curriculum reform in senior high schools, change the past that students did not pay enough attention to understanding the value of mathematics, understand the thinking methods of mathematics, enhance their self-confidence in learning, learn to communicate in mathematics, pay attention to the integration of knowledge and skills, processes and methods, emotional attitudes and values, pay attention to the requirements of the times and society for mathematics, pay attention to students' adaptability to society, and integrate knowledge learning, ability training and emotional formation to truly make students develop for life. Pay special attention to the following relations.
(A) correctly handle the relationship between "process" and "result"
In order to make students lay a good foundation, we should not only pay attention to the teaching process, but also pay attention to the teaching results. We should not let one tendency cover up another tendency, nor should we go from one extreme to the other. Because the result without process is the result without experience and deep understanding, and the process without pursuing the result is a process without value and significance.
First, try to reveal the essence of mathematics, return to the truth, and emphasize the real understanding and mastery of the basic concepts and methods of mathematics. In mathematics teaching, "reasoning is more important than reasoning". Through the analysis of typical examples, students are guided to experience the process of abstracting mathematical concepts from specific examples, so that students can understand the ins and outs of basic mathematical concepts and conclusions, thus understanding the thinking methods contained in them, pursuing the historical footprint of mathematical development, and transforming the artistic form of mathematics into an educational form that students can easily accept. For example, to understand the concept of derivative, students can experience the transition from average rate of change to instantaneous rate of change through examples. By calculating the instantaneous change rate, students can understand the actual background and significance of the concept of derivative and understand the idea and connotation of derivative. Some core concepts and basic ideas (such as function, vector method, space concept, combination of numbers and shapes, random concept, algorithm, etc.). ) should run through high school mathematics to help students deepen their understanding step by step. In particular, the thinking method contained in explicit knowledge, although tacit knowledge, is the "golden key" to open the treasure house of mathematics, so we must pay attention to revealing and summarizing it. The second is to pay attention to moderate normalization. Formalization is one of the basic characteristics of mathematics. In mathematics teaching, learning formal expression and application is also a basic requirement. For example, the application of some mathematical laws, formulas and conclusions should be mastered by students. This formalization is the regularity that students master by abstracting and summarizing after experiencing relevant mathematical concepts and thinking methods. If students only memorize formal expressions and ignore the understanding of the essence of mathematics, they will drown the lively mathematical thinking activities in the formalized ocean. Third, we should attach importance to thinking training and basic skills training. Choose appropriate forms to let students experience intuitive perception, observation and discovery, inductive analogy, spatial imagination, abstract generalization, operational solution, deductive proof and reflective construction in the learning process, so that the extensiveness, rigor, divergence, profundity, criticism and originality of thinking can be fully developed, thus forming rational thinking and learning critical thinking. At the same time, we should pay attention to the training of basic skills such as operation, drawing, reasoning and data processing to improve students' applied mathematics ability. Fourth, pay attention to the connection between knowledge and improve students' understanding of mathematics as a whole. Because the new curriculum is presented in the form of modules and special topics, special attention should be paid to the communication between the contents of each part. For example, in the teaching of solid geometry, we should pay attention to the use of vector method (algebraic method) to deal with related problems, pay attention to its geometric background and application in inequality teaching, strengthen the connection between vector and trigonometric constant deformation, vector and algebra, number and shape, and the infiltration and application of algorithm ideas in related content, so that students can have a better understanding of the results of mathematics learning.
(2) Correctly handle the relationship between "laying a solid foundation" and "striving for innovation"
Foundation and innovation are two indispensable aspects in the process of learning mathematics, and they are also fully emphasized in the curriculum standards. Some people think that these are two contradictory aspects, and cultivating innovative spirit will affect "double basics". In fact, this idea still stems from the incorrect understanding of "double base". From the perspective of social development, innovative spirit is one of the basic qualities that modern people must possess, and it is also the content of "two basics". We should stimulate students' innovative potential while laying a good foundation, and carry out the innovative spirit throughout. The two are not separated, but consistent.
Therefore, students must be provided with the space of "asking questions, exploring thinking and practical application". First of all, we should improve teaching and learning methods and advocate proactive and exploratory learning methods. Students' mathematics learning activities should not be limited to memorizing, imitating and accepting concepts, conclusions and skills, but should also advocate independent exploration, independent thinking, hands-on practice, cooperation and communication, reading and self-study and other ways to learn mathematics. Different teaching methods can be used for different contents. For example, through data collection, investigation, discussion and communication, we can give full play to students' initiative in learning and make the learning process an innovative process under the guidance of teachers. Although teachers' teaching is one of the important teaching methods, we should attach importance to students' subjective participation, including thinking participation and behavior participation. It is necessary to create appropriate problem situations and encourage students to discover mathematical laws and methods to solve problems. Second, we should pay attention to the cultivation of innovative thinking and mathematical application consciousness. In teaching, teachers should leave appropriate space and time for students to expand and extend according to different contents, objectives and students' actual situation, and further explore and study related issues. For example, the concept of inverse function, Euler polyhedron theorem, continued fraction and so on. Can be used as an extension and extension of the content. It is also necessary to carefully set questions to inspire students to think positively, so that students are often in a position of "jumping to pick peaches". At the same time, we should pay attention to cultivating students' application consciousness and practical ability, and tap teaching resources based on students' real life and social practice. On the one hand, introduce mathematical knowledge through rich examples. For example, at the beginning of each chapter, you can ask a practical question with a strong real life background, only asking and not answering, creating suspense and stimulating curiosity. In fact, abstract concepts such as function and derivative can be derived from examples. On the other hand, students should be guided to apply mathematical knowledge to find and solve practical problems, for example, using functions, statistics, derivatives and other knowledge to directly solve the maximum volume problem of stadiums and gymnasiums, commodity marketing strategies and so on. It is also necessary to guide students to find problems from actual situations and turn them into mathematical models through mathematical modeling activities, and try to solve problems with mathematical knowledge and methods, with emphasis on the application of logical knowledge, so that students can realize that mathematics is related to me and real life, and mathematics is useful. This can not only cultivate innovative consciousness, but also lay a solid foundation. Third, vigorously carry out mathematical inquiry activities. The problem is the core of mathematics. Teachers should always ask questions worth studying or exploring. Through the exploration of mathematical problems, the process of accepting mathematical learning is transformed into the process of exploring problems, so that the formation process of knowledge is paid attention to, and the learning based on imitation and memory becomes a vivid and personalized experience of solving problems, an experience of discovery and creation, and the tool function of mathematics and the thinking training function are unified in the process of solving problems. Turning knowledge into problems is an effective way to cultivate innovative spirit and lay a good foundation, which is easier to promote students' independent exploration and cooperative exchange and realize different development of different people in mathematics.
(3) Correctly handle the relationship between "laying a good foundation" and developing emotions and values.
There is also an important concept in the curriculum standard, which is to integrate emotions, attitudes and values into the curriculum. In fact, emotion and will play a dynamic role in people's growth and undertake tasks such as orientation, maintenance and adjustment. The Outline of Basic Education Curriculum Reform (for Trial Implementation) also clearly puts forward: "Change the tendency that the curriculum pays too much attention to imparting knowledge, and emphasize the formation of a proactive learning attitude, so that the process of acquiring basic knowledge and basic skills becomes the process of learning to learn and forming a correct world outlook at the same time." It can be seen that laying a good "double foundation" is completely consistent and complementary to stimulating learning interest, forming a proactive learning attitude, advocating the rational spirit of mathematical thinking, and establishing a dialectical materialist world outlook. The formation of students' learning emotions and correct values is also an integral part of the foundation, and knowledge and practice should be integrated in teaching.
First, let students fully appreciate the cultural value of mathematics. Mathematics is an important part of human culture. In teaching, students should be guided to understand the interaction between mathematics and the development of human society. For example, some historical events that play an important role in the development of mathematics are introduced in combination with the course content: Euclid's thinking method of establishing axiom system has a great influence on human rational thinking, mathematics and scientific development; Analytic geometry founded by Descartes, calculus founded by Newton and Leibniz, and their role in promoting the scientific society and the progress of human thought after the Renaissance; The appearance of computer and its role in social progress; Wait a minute. The second is to introduce mathematicians' innovative spirit and struggle history, fully display mathematicians' great personality and lofty spirit of dedication to truth, and set an example for learning. The third is to create a good mathematical situation and strive to create a successful environment for students. It is necessary to pay attention to the feasibility and stimulation of topic selection, design exercises with different requirements for different students, so that different students can learn different mathematics and valuable mathematics, guide students to overcome difficulties and have opportunities for success, thus establishing students' learning confidence. Fourth, strict requirements, using the scientific thought system contained in mathematics itself to guide students to actively explore, and develop the study habit of seeking truth from facts, being diligent and meticulous, bravely overcoming difficulties and persevering. It should be noted that the learning of mathematical culture and the cultivation of emotion should be based on the teaching content, which should be gradually infiltrated vividly, interestingly and naturally, so that students' knowledge and emotion can develop together in a subtle way.
Third, the new "double basics" new requirements for teachers
Obviously, the "double basics" under the curriculum standard has richer and more contemporary connotations. It is more difficult than before to lay the "double basics" well, and it also puts forward new requirements for teachers. Because teachers are the implementers of the new curriculum and an important force in the research, construction and resource development of the new curriculum, teachers are the key to the success of the "two basics". The author believes that as a teacher, we must pay attention to the following aspects.
(A) change ideas and establish new ideas
Through study, we can fully understand our role and function in the reform of mathematics curriculum and lay a good foundation. Teachers should not only be disseminators of knowledge, but also guides, organizers and collaborators of students' learning. According to the concept of "let different students learn different mathematics and get different development in mathematics", students are given room for development, and specific guidance is given according to their different levels, interests and development directions, so that knowledge and skills, processes and methods, emotional attitudes and values can be organically integrated, and students' foundation and quality can be fully developed.
(B) to strengthen knowledge accumulation
New standards and high requirements. For the first time, teachers are in the position chosen by students. They must re-examine their knowledge structure and teaching methods, study hard the new theories and knowledge of mathematics, grasp the academic frontier trends, broaden the knowledge of related disciplines, and realize the exchange and integration of multiple disciplines. At the same time, improve teaching methods, actively explore teaching methods suitable for senior high school students' mathematics learning, always maintain an attitude of research and innovation, and influence students with profound knowledge, solid basic knowledge and positive attitude towards life.
(3) Improve the evaluation method and establish a scientific evaluation mechanism.
Teachers' evaluation is a baton to students in a sense, which should be conducive to the development of students' "double basics" and the all-round growth of students. It is necessary to correctly and comprehensively evaluate students' basic knowledge and skills in mathematics, attach importance to understanding the essence of mathematics and mastering thinking methods, attach importance to the evaluation of students' learning process (including learning methods and attitudes), and attach importance to the evaluation of students' various mathematical abilities. We can also carry out diversified evaluation according to students' different choices to promote students' development.
In short, the new curriculum reform requires us to re-examine the connotation of "double basics" and seriously think about methods and ideas to lay a good foundation for students' lifelong development in the new period.