The so-called mathematical thought is the essential understanding of mathematical knowledge and methods and the rational understanding of mathematical laws. The so-called mathematical method is the fundamental procedure to solve mathematical problems and the concrete embodiment of mathematical thought. Mathematical thought is the soul of mathematics, and mathematical method is the behavior of mathematics. The process of using mathematical methods to solve problems is the process of accumulating perceptual knowledge. When the accumulation of this quantity reaches a certain procedure, it produces a qualitative leap and thus rises to mathematical thought. If mathematical knowledge is regarded as a magnificent building built by a clever blueprint, then mathematical methods are equivalent to architectural means, and this blueprint is equivalent to mathematical thoughts. Mathematical thought is the soul of mathematical method, and mathematical method is the manifestation and means of realization of mathematical thought. Therefore, people collectively call it mathematical thinking method. Therefore, in mathematics teaching, teachers should not only impart basic knowledge and skills, but also pay attention to the infiltration of mathematical thinking methods and cultivate students' mathematical thinking methods, which will have a far-reaching impact on students' future mathematics learning and application of mathematical knowledge. Paying attention to the infiltration of mathematical thinking methods from junior high school will lay a solid foundation for students' follow-up study and benefit them for life. Teaching that only pays attention to imparting superficial knowledge without infiltrating mathematical thinking methods is incomplete, which is not conducive to students' real understanding and mastery of what they have learned, and makes students' knowledge level stay in the primary stage forever, which is difficult to improve; On the other hand, if we simply emphasize mathematical ideas and methods while ignoring superficial knowledge, teaching will become a mere formality, and students will find it difficult to understand the true meaning of deep knowledge. Therefore, the teaching of mathematical thought should be integrated with the teaching of the whole surface knowledge.
First, the thinking methods that should be infiltrated in junior high school mathematics teaching
1, the idea of transformation
Conversion thought is the most basic and widely used mathematical thought. When studying and solving mathematical problems, we should understand and flexibly use the relationship between old and new knowledge, and with the help of some graphic properties, formulas or known conditions, turn the problems to be solved or difficult to solve into solved or relatively easy to solve problems in a certain way, and finally get the answer to the original problem. In the teaching of real number operation, solving equations (groups), polygon interior angle sum, geometric proof and so on. Students understand the transformed thinking method and consciously or unconsciously accept the transformed thinking. For example, on the basis of addition, the concept of reciprocal is used to reduce the law, so that addition and subtraction are unified and the concept of algebraic sum is obtained; On the basis of multiplication, the concept of reciprocal is used to simplify the division rules and unify the two reciprocal operations. Another example is to explore the nature of isosceles trapezoid. In addition to the axisymmetric method in textbooks, isosceles trapezoid is often transformed into parallelogram and triangle knowledge by making a waist parallel line, making the bottom high and extending at a point where the two waists intersect.
In addition, there are mutual infiltration and transformation among many knowledge: multivariate transformation into monism, high order transformation into low order, fractional transformation into algebraic expression, general three solutions into special triangles, polygons into triangles, algebraic solutions to geometric problems, and inequality knowledge to solve identity problems. ...
Step 2 discuss ideas by category
The idea of classified discussion refers to the mathematical idea that when solving mathematical problems, sometimes the problems to be studied and solved are divided into several different situations according to the characteristics and requirements of the problems and certain standards, and then they are studied and solved one by one according to each different situation. The thinking method of classified discussion widely exists in all knowledge points of junior high school mathematics. In teaching, a lot of complicated knowledge can be organized if the knowledge is properly classified.
For example, the definition of real numbers in textbooks is "rational numbers and irrational numbers are collectively called real numbers", which reveals the connotation and extension of real numbers, which in itself embodies the classified thinking method. So, after learning the concept of real numbers, you can classify them like this; Then when it comes to real numbers, you will think that it may be rational or irrational; When it comes to rational numbers, you will think that they may be integers or fractions.
For another example, the definition of absolute value of real numbers is also given by classification, and a = 0 is selected as the classification standard in this definition. In each category, the results do not contain absolute value symbols. Therefore, the definition also gives a way to get rid of the absolute value sign.
For another example, in the same circle, the circumferential angle of an arc is equal to half the central angle of the arc. In order to test this conjecture, the circle is often folded in half in teaching, so that the crease passes through the center of the circle and the vertex of the fillet. There are three possible situations: (1) the crease is one side of the fillet, (2) the crease is inside the fillet, and (3) the crease is outside the fillet. The verification should be divided into three situations, which actually embodies the thinking method of classified discussion.
Also, the exploration of triangle congruence identification method, the thinking question in the textbook: if two triangles have three parts (sides or angles) corresponding to each other, what are the possible situations? At the same time, the textbook also uses classified discussion to deal with several identification methods, from simple to complex, step by step, so that students can experience this way of thinking in teaching.
3. The combination of numbers and shapes.
Generally speaking, people call algebra "number" and geometry "shape". Numbers and shapes seem to be independent of each other, but in fact, under certain conditions, they can be transformed into graphic problems, and graphic problems can also be transformed into quantitative problems. The effective combination of quantitative relations and geometric figures will often make abstract problems intuitive, simplify complex problems and achieve the purpose of optimizing the way of solving problems.
The introduction of the number axis in the seventh grade textbook laid the foundation for the idea of combining numbers with shapes. The comparison of rational numbers, the geometric meaning of opposites, the geometric meaning of absolute values, and the graphic analysis in solving sequence equations fully show the power of the combination of numbers and shapes. This combination of abstraction and image can exercise students' thinking.
All grades make full use of the combination of numbers and shapes. For example, the positional relationship between a point and a circle can be determined by comparing the distance between the point and the center with the radius of the circle, the positional relationship between a straight line and a circle can be determined by comparing the distance from the center to the straight line with the radius of the circle, and the positional relationship between a circle and a circle can be determined by comparing the distance between two centers with the sum or difference of the radii of the two circles. Another example is the demonstration of the conclusion of Pythagorean theorem, the image of function and the nature of function, the approximate solution of binary linear equations is obtained by image, and the right triangle is obtained by trigonometric function. For another example, the addition rule of rational numbers, the multiplication rule and the determination of the solution set of inequality groups are all summarized by the number axis or other real graphs; In the teaching practice and exploration of travel problem, the method of line segment diagram is often used to guide students to analyze the quantitative relationship in the problem.
In mathematics teaching, the idea of combining numbers with shapes to help numbers has the advantage of presenting problems intuitively, which is conducive to deepening students' memory and understanding of knowledge; When solving mathematical problems, the combination of numbers and shapes is helpful for students to analyze the relationship between numbers in problems, enrich appearances, trigger associations, enlighten thinking, broaden their thinking, and quickly find solutions to problems, thus improving their ability to analyze and solve problems. Grasping the ideological teaching of the combination of numbers and shapes can not only improve students' ability to transform numbers and shapes, but also improve students' ability to transfer their thinking.
4. Overall thinking
The whole idea stands out in junior high school textbooks. For example, in real number operation, the number and the preceding "+,-"symbols are often regarded as a whole; For another example, using letters to represent numbers fully embodies the overall idea, that is, a letter not only represents a number, but also represents a series of numbers or a formula composed of many letters. Another example is that in algebraic expression operation, a formula can often be treated as a whole, such as: (a+b+c)2= [(a+b)+ c ]2, which is an excellent opportunity to cultivate students' good thinking quality and improve the efficiency of solving problems.
5. Mathematical modeling ideas
The idea of mathematical modeling refers to the mathematical thinking method of analyzing the quantitative relationship of problems, expressing practical problems in a mathematical way, establishing mathematical models, then solving them in a mathematical way and explaining practical problems according to the results. According to different practical problems, equations, inequalities, functions, geometry and other models can be established.
Example (20 10 Jiangxi province senior high school entrance examination) 25 shaver consists of a blade and a tool holder. Two manufacturers, A and B, produce old razors (blades can't be replaced) and new razors (blades can be replaced) respectively. Information about sales strategy and price is shown in the following table:
In a certain period of time, manufacturer A sold 8400 razors, while manufacturer B sold 50 times as many blades as the tool holders, and manufacturer B earned twice as much profit as manufacturer A. How many tool holders did manufacturer B sell during this period? How many blades?
6. Comparative thinking
The so-called comparison refers to distinguishing the similarities and differences in thinking between two or more similar research objects. Comparison is the basis of all understanding and thinking. With the deepening of learning, students have to master more and more knowledge, which requires students to be good at comparing the differences and connections between knowledge.
For example, in the teaching of factorization, by reviewing algebraic expression multiplication, let students compare the similarities and differences between the two operations, and make it clear that factorization and algebraic expression multiplication are identical deformation and reciprocal operations. For example, (a+b) (a-b) = A2-B2 is algebraic expression multiplication, and A2-B2 = (a+b) (a-b) is factorization. In the teaching of inequality solution, we can compare the solution of one-dimensional linear equation: the steps of removing denominator, removing brackets, moving terms, merging similar terms and changing the coefficient to 1 are the same. Of course, when the coefficient is 1, we need to compare the differences between them. For another example, congruent triangles is a special case of similar triangles when the similarity ratio is 1, and the similarity and congruence of two triangles have their specific internal relations. Therefore, the identification method of congruent triangles can be compared with that of similar triangles. For another example, axisymmetric figure, rotationally symmetric figure and centrally symmetric figure are concepts with different meanings, and the similarities and differences between them can be found through analogy, thus deepening the understanding of the essential attributes of these concepts.
Second, how to strengthen the infiltration of mathematical thinking methods in junior high school mathematics teaching
1.
Mathematical concepts, laws, formulas, properties and other knowledge are clearly written in the textbook, with a "shape", while mathematical thinking methods are implicit in the mathematical knowledge system, without a "shape", and are scattered in all chapters of the textbook systematically. Teachers talk as much as they want, which is very arbitrary. Because of the tight teaching time, they often squeeze it out as a "soft task". The requirement for students is to calculate as much as they can. Therefore, as a teacher, we should first renew our ideas, constantly improve our understanding of the importance of infiltrating mathematical thinking methods, integrate both mastering mathematical knowledge and infiltrating mathematical thinking methods into teaching purposes, and integrate the requirements of teaching mathematical thinking methods into lesson preparation. Secondly, we should study the teaching materials deeply and try our best to find out all kinds of factors that can penetrate mathematical thinking methods. For each chapter and section, we should consider how the specific content permeates mathematical thinking methods, which mathematical thinking methods permeate, how to permeate, and to what extent. It is necessary to have an overall design and put forward specific teaching requirements at different stages.
2. Grasp the feasibility of infiltration
The teaching of mathematical thinking method must be realized through specific teaching process. Therefore, we must grasp the opportunity of teaching mathematical thinking methods in the teaching process-concept formation, conclusion derivation, method thinking, thinking exploration, law revelation and other processes. At the same time, we should pay attention to the organic combination and natural infiltration in the teaching of mathematical thinking methods, consciously and imperceptibly inspire students to understand all kinds of mathematical thinking methods contained in mathematical knowledge, and avoid the counterproductive practices such as mechanically copying, generalizing and being divorced from reality.
3. Pay attention to gradual and repeated infiltration.
Mathematical thinking method is gradually accumulated and formed in the process of enlightening students' thinking. Therefore, in teaching, we should first emphasize "reflection" after solving problems. Because the mathematical thinking method refined in this process is easy for students to understand and accept. Secondly, we should pay attention to the long-term nature of infiltration. It should be noted that the infiltration of students' mathematical thinking methods can not see the improvement of students' mathematical ability overnight, but has a process. Mathematical thinking methods must be trained step by step and repeatedly, so that students can really understand.
In short, in mathematics teaching, as long as we grasp the above typical mathematical ideas and pay attention to the infiltration process, according to the content of teaching materials and students' cognitive level, we will certainly improve students' learning efficiency and mathematical ability from the first year of high school.