Functions images embodied in dialectical point of view _318

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Functions images embodied in dialectical point of view
 
 
In the third function and its image in the algebra, contains extremely rich dialectical point of view. This chapter is the biggest characteristic of the teaching content mathematics, with the variables, differentiation and integration will be necessary once the. Deepening the basis of previously learned knowledge and understanding have to improve the mathematical ability to form a movement, change, contact the consciousness, but also to train students in a more natural outlook of dialectical materialism.   one, constant and variable   dialectics that everything in the world are interrelated, movement, change and development. Constant, as opposed to a process or in terms of another variable. Constant is not absolute. Because the motion of matter is absolute rest is relative, so the dynamic is changing things. If so, is there a relatively constant, absolute constant does not exist. Therefore, in the teaching process, to help students understand the constant dialectical relationship with the variables that may wish to take the following example. (1) uniform linear motion, the speed is constant, time and distance are the variables; and the actual movement in the process. Absolute uniform motion is not. For example, a student to ride home is easy to see the daily movement of the process, they will inevitably acceleration, deceleration, braking and so on. (2) cinema box office statistics, screening and seat in a particular category, the fare is a constant, and the ticket number of sheets and income are variables; but relative to a longer time interval, since the content of the show , species, different grades, the fare is still a variable. (3) days or for several days in a measurement of a student's height, can be approximated as constant; but this student's height, if you look at from a longer period of time, it is variables.  Practice shows that constants and variables to make students understand the dialectical relationship, they must form, multi-angle, multi-level to be explained.   Second, exercise and rest   according to the objective laws of human understanding of things and youth development practices and knowledge, we can combine materials the specific teaching content, to guide students to understand things step by step the absolute and relative movement of the dialectical relationship between the static.   example, we can guide the students to see from the textbooks, exercise books or on the board in the y = x drawn image to think about: the image is stationary on the surface , but in the list, described the process point to see the connection is movement, change. Further excavation can be found: the image painted on the surface is complete, in fact, is not complete, because it can be infinitely extended to the two parties, namely, constant movement, development and change, the image will forever be a function of draw can be localized, it can only be a function image of a symbol; the same time some of the examples also reflects the dialectical unity with the whole.   III  content and form of materials under the current system, who started last semester, students learn about the concept of the equation would think, of the form y = 2x +1 in the formula represents a binary one equation; grade students new to the concept of a function, y = 2x +1 will be considered to represent a time function; when the students describe the general method of images delineate the function y = 2x +1, after the image, but also recognize that y = 2x +1 can also be said that a straight line. From a philosophical point of view, y = 2x +1 nature of a class of things that link, its content is extremely rich, and rich content in the form of expression which is the same. This is that the same thing under different external conditions can have a variety of external manifestations of the same external form of the different nature of that content. With the increase of knowledge and understanding of student capabilities, their understanding of the nature of things will be gradually increased from the emotional as rational.   Fourth, special and general   dialectics that the general resides in particularity. Materials involved in the special and general content of at least the following aspects: (1) y = kx and y = kx + b; (2) y = ax2 and y = ax2 + k; (3) y = ax2 and y = a (xh) 2; (4) y = ax2 and y = ax2 + bx + c. The relationship between them, are typical of the relationship between special and general, and this relationship is a dialectical unity. To help students understand the essential nature of things, always introduce a simple teaching materials,You are not allowed to view links. Register or Login, special content, and then gradually extended, and gradually deepened to more complex, more general content to guide students to gradually understand the essential nature of things, master understanding of the law of things.   five, phenomenon and essence  in the physical world, without a certain phenomenon, we can not show the nature of things, but its nature is often resides in the phenomenon of being. Of course, individual cases can reveal the nature of things not necessarily because the nature of the phenomenon of the apartment owned by a number of similar. This will be true in mathematics.   example, in the first grade, students can successfully determine the solution set of equations for the empty set, and relative to the understanding of a parallel line, naturally there is no intersection A quadratic equation x2 +2 x +3 = 0 Why is there no real solutions? Function y = x2 +2 x +3 x-axis image and why there is no point of intersection? Function y = x2 +2 x +3 What is the minimum? Students To study the most general function y = x2 +2 x +3 minimum, an exercise in student Deep understanding of the nature of the phenomenon to the process. These problems, the equation has no real roots, or images with x-axis does not intersect or vertices in the x-axis side, are all phenomena, the nature of the problem, it is the state constraints for such problems. Another example of how to solve x-3> 0, x-3 = 0, x-3 <0, also belong to the phenomenon, and accurate understanding of the function y = x-3 properties, is understanding of the nature of things.  As viewed from the outside, y = a1x2, y = a2x2 + k, y = a3 (xh) 2, y = a4 (xh) 2 + k, y = a5x2 + bx + c, they are different; but ai (i = 1,2, ..., 5) non-zero real constant, b, c, h, k are real constants, their essential features to expose, which was Now we are actually in front of the same type of thing: all on behalf of a parabola; In particular, when a1 = a2 = a3 = a4 = a5 ≠ 0, they are common to more thoroughly exposed, after the four may be represented by the parabola y = a1x2 by be appropriate to change position, while the opening direction, the size did not change.

VI  concrete and abstract theory of modern cognitive science tells us that the essential attribute of human understanding of things, is the phenomenon to nature, from concrete to abstract, progressive approach to the evolutionary process. Perceptions often returning to the thinking of some specific practice; and rational knowledge is returning to the initial understanding of the process of generalization and abstraction, so as to achieve the understanding of the essential nature of things. Therefore, only the specific perceptual knowledge from the development of abstract rational knowledge only after easily incorporated into existing cognitive structures, it can be transformed into the ability to use in order to provide higher level of abstraction as the basis and guarantee. We can read the materials found, whether it is directly proportional to the function, a function, quadratic function of, or the inverse function of the image and nature of the discussion is to start gradually from concrete to abstract discussion and argument, thus deepening understanding of this knowledge. To make the students understand not limited to the specific, in making it gradually increased as the abstract, teaching materials and speak well of each specific typical example, I always come to a which captures the essential link. It is this To help students overcome cognitive obstacles, we should give students a wealth of emotional material, thus producing a rich perceptual, and then gradually rise to rational understanding.  Seven quantitative and qualitative change  quantitative and qualitative change in this chapter reflect the views of the contents of many examples, giving students a deep understanding of the content is very difficult and therefore we should be teaching step by step guide, bit by bit penetration, and then to systematically promote the understanding of the content. (1) For a function y = kx + b, if changed from k ≠ 0 k = 0, how? (2) the quadratic function y = ax2 + bx + c, the provisions of a ≠ 0; if so a = 0, the situation? (3) inverse function y = in the range of the independent variable x is x ≠ 0; if x = 0, or y = 0, how will? (4) y = kx + b, from k> 0 becomes k  0 becomes Δ = 0 or Δ  -, the change trend of the change accordingly?   the like, are the accumulation of quantitative to a certain extent led to a qualitative change in the example.  VIII   finite and the infinite number of things or in the performance of concrete is always limited, so we have this concept can be extremely poor or easy to understand, or be able to fully grasp; and infinite is the abstract, it is a sport unlimited extension of the process is a material change in trend, is an abstract concept, to be repeated before a certain level of penetration understanding.  (1) students , that is, to use a limited part of the (2) list, describe points, connections, draw a parabola, the function obviously only draw the image of a connection is limited to the infinite. (3) inverse function of the image in the painting when, on the finite and the infinite, the limit of thinking is reflected more fully, for example, a textbook example y = observed image, when x (or y) the greater the absolute value (or smaller) when, y (or x) the absolute value of change? What is How many points within? Cartesian coordinate plane, how many points? axis between any two points in the number of points? to any point within the coordinate plane P (a, b) a circle, any small positive number r is radius for the round, round, how many points? circle how many points? round things, the subtle influence on the understanding of the number of infinite penetration, limit other point of view.  Nine, discrete and continuous  discrete and continuous two aspects of a contradiction, but in the list - description of points - the process of connecting in discrete and continuous connection to get a uniform. Such as textbooks, drawing y = x and y = x2 images are used from simple to complex, from specific to general, from discrete to continuous way, a manifestation of this unity of opposites.  Careful analysis of the materials, not difficult to find In addition to the main points have been described above, content, at least the micro and macro, straight and curve, precision and approximation, part and whole, absolute and relative, subjective and objective dialectical unity and so on. Due to space limitations, not going to repeat them.  To help students develop dialectical view of the world, we should be based on the teaching content related to textbooks, combined with the level of awareness of students, purposeful, planned, systematic, focused and organized teaching content, easily accepted by students of education, teaching methods, adequate penetration, the system forward, when the penetration to a certain extent, and then sort out timely and appropriate to carry out general and abstract; over time, the contents of these teaching system in the minds of students and went to take root deeply. This curriculum provides training in the task of dialectical materialism can be successfully completed.