[Keywords:] new curriculum standards, mathematical concepts, philosophical thinking, effective way variants
For a long time, due to the influence of exam-oriented education, many teachers pay more attention to solving problems than concepts, which leads to a serious disconnect between mathematical concepts and solving problems. Students are vague about concepts and lack knowledge, so they can't understand and use them well. Mathematics classroom has become a place where teachers train students to solve problems, and students have become problem-solving machines. This situation greatly affects the quality of teaching, and students are also deeply involved in the sea of questions, and their learning efficiency is very low; More seriously, it will hinder the development of students' thinking and the improvement of their ability, which is contrary to the cultivation of students' inquiry ability and innovative spirit advocated by the new curriculum.
1, the requirements of the new curriculum standard for mathematics concept teaching
The new curriculum standard of high school mathematics clearly points out: let students acquire the necessary basic knowledge and skills of mathematics and understand the basic concepts and essence of mathematics. In high school mathematics teaching, it is necessary to strengthen the understanding and mastery of basic concepts and ideas, understand their background, application and role in subsequent learning, and put some core concepts and basic ideas into high school mathematics teaching to help students gradually deepen their understanding and appreciate their mathematical ideas and methods. Because mathematics is highly abstract, we should pay attention to the context of basic concepts, explore the connotation and extension of key and core concepts, guide students to experience the process of abstracting mathematical concepts from concrete examples, and gradually understand the essence of concepts in preliminary application.
2. The requirements of mathematics concept teaching for teachers under the new curriculum standards.
Mathematical concept is the cornerstone of mathematical building, and mathematics is a theoretical system composed of many interrelated concepts through logical reasoning. As an important part of mathematics teaching, concept teaching should also conform to the trend of educational reform and innovate constantly.
Under the new curriculum standards, teachers should update their teaching concepts and attach importance to conceptual teaching. In the specific teaching process, students should be made aware of the universality of concepts and the mathematical ideas infiltrated in concepts, stimulate their interest in learning mathematics, and improve their ability to think, summarize and apply concepts to solve problems. The teaching of mathematical concepts does not lie in how thoroughly the teacher explains the concepts, nor does it lie in imposing the concepts on students. Instead, it inspires, guides and encourages students to actively explore problems according to the background of the concept itself and the knowledge that students have mastered, so as to learn and construct mathematical concepts in the exploration activities. Therefore, the teaching of mathematical concepts under the new curriculum requires teachers to position themselves reasonably and realize the renewal and upgrading of their roles.
In the traditional teaching of mathematical concepts, teachers often only pay attention to the teaching of concepts, ignoring the background introduction of concepts and the analysis of students' cognitive structure, which can't make students understand concepts from many sides and angles and correctly analyze the essential and non-essential attributes of mathematical concepts. This kind of teaching will only make students form some isolated knowledge blocks in their minds, which is not conducive to students' comprehensive use of knowledge to analyze and solve problems, and violates the learning theory of constructivism. If the teaching of mathematical concept course is regarded as a kind of film cultural activity, then the teacher is not only the investment promoter of the concept script, but also plays the role of screenwriter, director and film critic before and after the concept is formed.
3. Thinking about concept classroom teaching based on classroom process design.
The process of concept class returning to the classroom is nothing more than three stages: concept introduction stage, concept exploration stage and concept application stage. How to improve the effectiveness of mathematical concept course? The author thinks according to the three stages of the classroom process.
(1) Concept introduction stage: the question should be put forward with practical significance, which can arouse students' great interest, touch students' observation nerves and get close to the theme. Through the intuitive feeling of contradiction, real life or graphics, give students appropriate perceptual knowledge, pave the way for breaking through difficulties, and then naturally introduce concepts.
There are two ways to introduce new concepts in middle school mathematics teaching: first, introduce them with practical examples or objects and models, so that students can understand the research object from perceptual to rational, gradually understand its essential attributes and establish new concepts. Especially in the teaching of analytic geometry and solid geometry concepts, such as teaching the concept of "cylinder, cone and platform", let students observe related objects, figures and models first, and then introduce concepts on the basis of full perceptual knowledge. Secondly, it is also an effective method to introduce concepts from the internal development needs of mathematics. For example, the introduction of concepts such as "imaginary number" and "dihedral angle". Third, the extension or deformation of the old concept leads to the emergence of new concepts. Such as "the modulus of vector", "the distance formula between two points" and "the inclination angle of a straight line".
(2) Concept exploration stage: Explore concepts, go deep step by step, mobilize students, discuss in groups and think positively. Enlighten and guide students in the inspection, keep abreast of students' trends, help students remember and understand, and form concepts.
The teaching of new mathematical concepts must seriously explore concepts, clarify the connotation and extension of mathematical concepts, and communicate the internal relations of knowledge. What are the terms and conditions in the concept? Is there anything confusing compared with other concepts? How do they relate to what they have learned in the past? What exactly do these terms and conditions mean? How should we understand these differences? Can these concepts be extended and deformed? This is what teachers should focus on.
Teachers should use various means in time to help students deepen their understanding of concepts. For example, students can retell concepts, give some relevant examples to help students grasp the connotation and extension of concepts, and compare them with some related concepts to find out their connections and differences. Such as permutation and combination, exponent and logarithm, trigonometric function and inverse trigonometric function, can all get good results through comparison. You can also use some idioms such as three-character formula and four-character formula to help you remember, such as the inductive formula of trigonometric function, "odd even, sign according to quadrant" and so on.
(3) Concept application stage: After the students know and form the concept, it is an essential link to consolidate the concept. By selecting examples, designing clever questions and strengthening exercises, we can consolidate and apply concepts, so that students can finally master mathematical thinking methods through mastering and applying concepts.
The main means of consolidation is to practice more and use more. Only in this way can we communicate the memory relationship among concepts, theorems, laws, properties and formulas. For example, after learning the concept of "the first definition and the second definition of ellipse", you can practice with examples and consolidate the original concept by solving problems. These exercises can be divided into two steps: first, start with basic exercises to help students get familiar with and master new concepts and knowledge. After mastering the basic content, design some small turning points, small changes and small comprehensive topics according to the actual situation of the class students, so that students can use their knowledge flexibly to solve problems.
4. Thinking about the teaching of mathematical concepts based on philosophical thinking.
4. 1 uses the view of universal connection.
Materialist dialectics holds that everything in the world is universally related. Mathematical concepts are no exception. Many concepts in mathematics are closely related, such as plane angle and space angle, equation and inequality, mapping and function, opposite events and mutually exclusive events. In teaching, we should be good at discovering and analyzing their connections and differences, which will help students grasp the essence of concepts.
4.2 Use contradictory views
Materialist dialectics holds that contradiction is the motive force and internal source of the development of everything. Similarly, mathematical concepts are also produced and developed in the process of constantly producing and solving contradictions. In the teaching of mathematical concepts, we should be good at finding the logical connection point between old and new concepts, setting up effective problem situations, causing conflicts between students and original knowledge, and stimulating students' enthusiasm.
For example, the concept of function, defined in junior high school, is from the perspective of motion change, and that function is the process in which one variable changes with another. Is that a function? Contradictions have emerged, and teachers can define the functions of high schools accordingly.
4.3 the use of quantitative change and qualitative change point of view
Materialist dialectics holds that any objective thing is an organic unity of quality and quantity, and the development process of things is a process from quantitative change to qualitative change. In the teaching of mathematical concepts, we should consciously infiltrate the dialectical thought of quantitative change and qualitative change.
For example, the tangent of a curve is the limit state of secant, and changing from secant to tangent is the process of quantitative change causing qualitative change. For another example, the quantitative change of eccentricity of conic curve causes the qualitative change of image from ellipse, parabola and hyperbola to two intersecting straight lines.
In teaching, understanding and guiding the teaching of mathematical concepts from the point of view that quantity change causes qualitative change can better touch the essence of concepts.
5. An effective teaching method of mathematical concepts.
5. 1 Effective variant teaching of mathematical concepts
Variant refers to the change form of a certain paradigm (such as basic knowledge, typical problems, ways of thinking, etc.). ), appropriately changed the problem situation, changed the thinking angle, and showed the essential attributes of the concept. Variant teaching is a process of using variant means to help achieve teaching objectives. Variant teaching can not only enhance students' perceptual experience and improve the accuracy of knowledge understanding, but also help to cultivate divergent thinking ability and improve the flexibility of thinking.
The introduction of concepts can be replaced by mathematics. When teaching new concepts, we should restore concepts to objective reality, introduce examples, models or existing experiences, and transplant the essential attributes of concepts through variants, so as to mathematize actual phenomena, so as to show the process of concept formation and promote the formation of students' concepts.
The differentiation of concepts can be changed in teaching. In view of the analysis of the connotation and extension design of the concept, through the discussion and solution of these problems by students, the purpose of clarifying the essence of the concept and deepening the understanding of the concept is achieved. For example, when learning the concept of function, students often mistakenly think that "only variables change with the changes of variables is a new function". Taking the non-essential attribute "change with the change of" as the essential attribute expands the connotation of the concept. At this time, we can give a positive example. When taking any non-zero real number, the unique value of y is 1, which can correct students' misunderstanding.
The application of concepts can also be changed to teaching. There are many examples and exercises of concepts in the textbook. If we can change the conditions and conclusions of the problem, we can more clearly define the application scope and relationship of related mathematical concepts. For example, after learning the concept of trigonometric function, you can design the following problem groups for students to explore. What are the domains of (1),, and? ② What are the symbols of trigonometric function in each quadrant? In this way, students can use newly acquired concepts in new problem situations, which will undoubtedly further deepen their understanding of concepts.
5.2 Effective experimental teaching of mathematical concepts
The concept of geometry, especially solid geometry, is abstract and difficult to understand. We can turn abstraction into concreteness by designing mathematical experiments.
For example, in the teaching of straight lines in different planes, we can design an experiment to demonstrate the three positional relationships of two straight lines in space-parallel, intersecting and different planes with three-color ribbons in the corner of the classroom.
For example, in the teaching of out-of-plane straight lines, the following experiments can be designed: let two students each take a pen, first make it vertical, and then find out the intersection angle, and then one of them moves his pen in parallel with the same angle, so the positional relationship after translation is called out-of-plane straight line vertical. That is, two straight lines in space do not intersect and form each other. This positional relationship is called that two straight lines in different planes are perpendicular.
5.3 Effective heuristic teaching of mathematical concepts
In concept teaching, teachers should provide opportunities, create situations, be good at asking questions, inspire students to think actively, and gradually cultivate students' ability of independent thinking and independent learning.
For example, when learning geometric series, we can design enlightening thinking questions, start students to observe, summarize and generalize the concept of geometric series independently, put the mathematical thought of analogy into practice, guide students to carry out conceptual analogy, connotation comparison, extension analogy, structural analogy of function formula and solution analogy in conceptual application, so that students can learn, understand and master geometric series and related concepts in analogy and independent exploration.
6. Concluding remarks
In a word, concept is the most basic way of thinking, and concept teaching and students' learning are the basis of learning mathematics, which is worth studying. Therefore, in the teaching of mathematical concepts in middle schools, only by focusing on students' reality and the specific characteristics of concepts, paying attention to introduction, strengthening analysis and training, supplemented by flexible and diverse teaching methods, can the quality of mathematical teaching be effectively improved.