About strengthening Shuxingjiege improve problem-solving ability

I. Introduction
Engels said: "Mathematics is the relationship between the amount of real world science and space in the form. Contradictory unity of the two studies in mathematical object "number" and "shape" the internal factors of the development of mathematics. The Combination is a thread throughout the development of mathematics, mathematics in practice more extensive and far-reaching. On the one hand, by virtue of the nature of the graphics to visualize abstract mathematical concepts and quantitative relationships, simplistic, giving the intuitive sense, on the other hand, the graphics problems into algebraic problems can get accurate conclusions. "Number" and "shape" of information conversion, mutual penetration, not only to make problem solving a simple and neat, but also to develop problem-solving ideas, has opened up an important way to study and explore mathematical problems. Shuxingjiege is connected to the "number" and "shape" of the "bridge", it is not only an important method of problem-solving, is an important mathematical ideas. High school mathematics learning the thought Shuxingjiege throughout.

Second, the purpose and significance of the study

The number is a form of abstraction, the shape is a visual representation of the number. Professor Hua said: "less intuitive when the shape of the number of missing, the form of a small number of difficult nuanced. Sorts of good Shuxingjiege, every crack separation MasterCard non." Shuxingjiege is to make full use of rigorous form and intuitive, abstract mathematical language combined with intuitive graphical language, the combination of abstract thinking and the thinking in images, graphic description, and algebraic argument to study and solve math problems, a mathematical way of thinking. The Combination of thinking, and its essence is to combine abstract mathematical language with intuitive image, the key is the mutual conversion between algebra problems and graphics, it can make the algebra problem geometry, the geometry issues algebra of.

Shuxingjiege way of thinking is the essence of one of the middle school math basics, is many knowledge into the ability to "bridge". High school mathematics teaching abstract problem students often find it difficult to understand, if teachers flexibility to guide students Shuxingjiege, into an intuitive, easy to perceive the problem, students easy to understand, and be able to solve the problem, to obtain a successful experience, enhance students' confidence in learning mathematics. Especially for the more difficult problems, the students, if resolved independently or teacher inspired and guided problem solving, the mood is more pleasant, so that it is easy to stimulate the enthusiasm, interest and enthusiasm of the students' learning of mathematics. At the same time, once the students master the Shuxingjiege law and continue to try and use, many problems can be solved.

Third, Shuxingjiege improve students' ability to solve problems

As a mathematical way of thinking, Shuxingjiege applications generally can be divided into two situations: either by means of a number of accuracy to clarify some of the properties of the form, or by means of shaped geometric intuition to clarify between the number of a certain relationship, i.e. Shuxingjiege includes two aspects: the first case is "In the number of solution-shaped", while the second case is "to form co-number". Which focus of Shuxingjiege is research to form the number of help. "
Based on the intrinsic link between the conditions and conclusions of the mathematical problem, it is necessary to analyze the algebraic sense, they reveal the geometric intuition, to accurately characterize the spatial form of the number of relevant visually clever, harmoniously together, and take full advantage of this Shuxingjiege looking for problem-solving ideas, anything easy, simplify, and thus be successfully resolved the problem of

("In-shaped co-number"

Review: the use of the number-shaped knot thinking not only intuitive and easy to find problem solving ways, but also to avoid the complex calculations and reasoning, and greatly simplifies the problem-solving process. In the solution of multiple-choice, fill-in more superior, the need to cultivate this ideology, in order to develop their own
Thinking vision.

Reviews: Shuxingjiege thought can make some abstract mathematical problem intuitive, vivid abstract thinking can change the image of thinking helps to grasp math the nature of the problem, to simplify the calculation.

Comment: Many of the functions of most value, there is a geometric background, with the form of intuitive problem solving is an important method of seeking problem-solving ideas, to the problem through a graphical geometric intuitive description, identify problems in Shuxingjiege logical relations, inspired thinking, problem Solving.

Review: vector has a good set of arithmetic nature, through the establishment of a Cartesian coordinate system, the geometry of nature can be transformed into vector operations, becomes abstract logical reasoning for vector operations, with the accurate specification to clarify the three-dimensional geometric rigor the properties, both to simplify the difficult problem of spatial imagination, especially simple.

Using Shuxingjiege ideological analysis and problem-solving, to pay attention to three things: first thoroughly understand the geometric meaning of some of the concepts and operation as well as the curve of the algebraic features analysis of its geometric meaning both on the the math topics conditions and conclusions and analysis Algebra is appropriate to set parameters, with reasonable parameters, build relationships, and by the number of Si-shaped to form want the number, good number of shape transformation, the third is the right to determine the parameter ranges.

Mathematics teaching penetrate Shuxingjiege thinking

Shuxingjiege is one of the important thinking of new high school math curriculum penetration. The contents of the new textbooks good training and development of students Shuxingjiege thinking. Penetration of this way of thinking in the textbook on developing students' problem-solving ideas, looking for the best problem-solving method has the role of guiding the correct analysis of the problem, compared reasonable Lenovo, and gradually form a correct view of the problem-solving, but also of abstraction given visualize understanding and memory, can guide the students in learning mathematical cognitive ability, and enhance the understanding of the real world, thereby enhancing the mathematical literacy, and constantly improve themselves.

Already full implementation of the new curriculum of teaching content hierarchy teach the new curriculum syllabus requirements and knowledge point of view, Shuxingjiege method of teaching through three main stages:
The first stage is the number of conformal maps, it is Shuxingjiege foundation, through the usual concept of teaching gradually penetrate students through learning, training, experience, and gradually comprehend and master. On one hand, the point on the real number corresponding with the number of axis points, flat with corresponding ordered pairs of real numbers, corresponding to the function and image corresponding to the curve with equation, etc., as well as the background of geometric elements and geometric conditions established concepts such as vectors, trigonometric functions, etc., to create conditions for Shuxingjiege provide theoretical support. On the other hand, the high school math concepts with strong abstraction, generality, students have a greater difficulty in understanding. Can make use of the geometric shape to achieve the purpose of helping students understand intuitive. For example, the function and image combine functional relationship expressed by geometric methods to help students understand the function of abstraction.

The second stage is the number of shape transformation, it reflects the number and shape relations in the problem-solving process, how as a method to be used. The mathematical problem is a prerequisite for carrying out mathematical thinking, problem-solving process, in essence, is a process of mental training. We can Shuxingjiege penetration in the problem solving process, mainly in the following three aspects:

(2 number to help the argument of the body will be the number of simple problem solving

(3 Shuxingjiege experience the unity of the two.
The third stage is the number-shaped division of labor, which is the application Shuxingjiege thinking as a strategy to solve the problem. For example, the third year review, the focus on creation of Shuxingjiege thinking topics, in order to achieve the purpose of the system to consolidate.

Throughout the years of the College Entrance Examination, the clever use Shuxingjiege the thinking methods to solve abstract mathematical problems, tend to do more with less. Therefore, the need to strengthen the teaching of high school mathematics Shuxingjiege to improve the quality of students' mathematical and problem-solving ability.