Common Mathematical Thinking Methods: Classification and Integration
When solving problems, we often encounter such a situation. After solving a certain step, it is impossible to continue with a unified method and formula, because the problem studied at this time contains many situations, and it must be correctly divided into several sub-regions within the total area given by conditions, and then solved in each sub-region. When solving problems by classification, we must also put them together, because we are studying the whole problem after all. This is ... It is not only the main process of solving problems with the idea of classification and integration, but also the essential attribute of this thinking method.
The college entrance examination puts the idea of classification and integration in a more important position and focuses on answering questions. Candidates are required to know which questions need to be classified, why and how to classify them, and how to learn and finally integrate them after classification. Pay special attention to the reasons for classification, we must be quite familiar with it. Some concepts are defined by classification, such as the concept of absolute value, the division of odd and even integers, and some algorithms and formulas are given by classification. For example, the summation formula of proportional series is divided into q= 1 and q? In the case of 1, the monotonicity of logarithmic function is divided into A >；; 1,0
The college entrance examination often focuses on analytic formulas with parameters, including function problems, sequence problems and analytic geometry problems. In addition, the problem of permutation and combination and probability statistics also examines the idea of classification and integration. With the implementation of the new curriculum in the national college entrance examination, it is one of the focuses of the college entrance examination proposition in the next few years to examine the idea of classification and integration in the new content.
Common Mathematical Thinking Methods: Functions and Equations
The famous mathematician Klein said? The important thing that ordinary educatees should learn in math class is to think with variables and functions? . A student who only studies the knowledge of functions is often passive in solving problems. Only by establishing the thought of function can he actively think about some problems.
Function is the backbone of senior high school algebra, and the idea of function runs through all the contents of senior high school algebra. Functional thought is the abstraction, generalization and refinement of functional content at a higher level, and it is to consider, study and solve problems from the internal relationship and overall perspective of functional parts.
The idea of equation is to highlight the equal relationship between known quantity and unknown quantity, and to achieve the purpose of evaluation by setting unknown quantity, listing equation or equation, solving equation or equation, etc. It is the basic idea to solve all kinds of calculation problems and the foundation of calculation ability.
Functions, equations and inequalities are related to each other through the function value being equal to zero, greater than zero or less than zero, and there are both differences and connections between them. The thought of function and equation is not only the embodiment of function thought and equation thought, but also the embodiment of their comprehensive application, and it is the basic mathematical thought in the process of studying variables and functions, equality and inequality.
The college entrance examination focuses on the thought of function and equation, and examines the basic application of function and equation with multiple-choice questions and fill-in-the-blank questions. In solving problems, it comprehensively examines the relationship between thinking methods and related abilities from a deeper level at the intersection of knowledge networks.
Learn to think about these questions when solving problems: (1) Do we need to regard letters as variables? (2) Is it necessary to regard algebraic expressions as functions? If it is a function, what are its properties? (3) Is it necessary to construct a function to turn a seemingly non-functional problem into a functional problem? (4) Can an equation be transformed into an equation? What are the requirements for the root of this equation?
Common mathematical thinking methods: special and general
From special to general, from general to special, is one of the basic ways for people to know the world. Mathematical research is no exception. The basic cognitive process of studying mathematical problems from special to general and from general to special is the special and general thinking in mathematical research.
When we study formulas, theorems and laws, we often start from the special and summarize them. After proof, we use them to solve related mathematical problems. Induction and deduction, which are often used in mathematics, are the concentrated expression of special ideas and general ideas. By analyzing the college entrance examination questions over the years, there are many questions that examine special and general ideas. Some of them use general induction to guess, some use special functions and special sequences to find special points, determine special positions, and use special values and special equations to study and solve general problems, abstract problems and motion changes. With the comprehensive promotion of new teaching materials, the college entrance examination is bound to become the direction of proposition reform in the future, taking new content as the material and highlighting special and general ideas.
Common mathematical thinking methods: finite and infinite
Finite and infinite are not new things. Although the mathematics we started to learn is limited in teaching, it also contains infinite elements, but it has not been studied in depth. In the process of learning numbers and their operations, the learning of natural numbers, integers, rational numbers, real numbers and complex numbers is a finite number of operations, but in fact the number of elements in each number set is infinite. In analytic geometry, I have also studied the asymptote of parabola, and the idea of limit has begun to be reflected in it. The limit of sequence and function embodies the idea of finiteness and infinity. Solving mathematical problems with the idea of limit, it is obvious that the volume and surface area of a sphere in solid geometry are solved by infinite division. In fact, it is divided by a limited number of times, and then summed to find the limit. This is a typical application of finite and infinite thoughts.
Function is a description of dynamic things that change in motion, which embodies the important role of variable mathematics in studying objective things. Derivative is a description of the change speed of things, which can further deal with and solve practical problems such as the increase and decrease of functions, maximum, minimum, maximum and minimum, and is a powerful tool for studying the change rate and optimization of objective things.
The examination of finite and infinite thoughts in the college entrance examination has just begun, and it is often in the process of examining other mathematical thinking methods to examine finite and infinite thoughts at the same time. For example, when inductive thinking is used from special to general, it contains finite and infinite thoughts; When using mathematical induction to prove, solve infinite problems, embody finite and infinite ideas, and so on. With the gradual deepening of the examination of new content, it is necessary to strengthen the examination of limited and infinite thoughts and design novel test questions that highlight limited and infinite thoughts.
Common Mathematical Thinking Methods: Possibility and Necessity
Random phenomena have two basic characteristics. One is the randomness of the results, that is, the results obtained by repeating the same experiment are different, so that it is impossible to predict the results of the experiment before the experiment; The second is the stability of frequency, that is, the frequency of each test result in a large number of repeated tests? Stable? Close to a constant. To understand a random phenomenon, we must know all the possible results and the probability of each result. Only by knowing these two points can we say that we have studied this random phenomenon clearly. Probability studies random phenomena, and the research process is in? Accidental? Looking for it? Inevitably? And then use it? Inevitably? To solve the law? Accidental? The mathematical thought embodied in this problem is the thought of possibility and inevitability.
With the popularization of new textbooks, the examination of the probability content of college entrance examination has been placed in an important position. By focusing on the probability of equal possibility events, the probability of mutually exclusive events's occurrence, the probability of mutually independent events happening at the same time, the probability of N independent repeated experiments just happening, the distribution list of random events and mathematical expectations, the basic concepts and methods are investigated, and the dialectical relationship between possibility and inevitability in solving practical application problems is investigated.
Probability problems, no matter what type they belong to, are all studied in random events. Possibility? With what? Inevitably? Dialectical relationship, in? Possibility? Looking for it? Inevitably? Law.
Common Mathematical Thinking Methods: Conversion and Transformation
Through observation, analysis, analogy and association, the idea of transforming an unknown solution or difficult problem into a solved or easy-to-solve problem within the scope of known knowledge is called transformation thought. The essence of transformation thought is to reveal the connection and realize transformation.
Except for extremely simple mathematical problems, the solution of each mathematical problem is realized by transforming it into a known problem. In this sense, solving mathematical problems is the transformation process from the unknown to the known. The idea of transformation is the fundamental idea of solving mathematical problems, and the process of solving problems is actually a gradual transformation process. There are many transformations in mathematics, from unknown to known, from complex problem to simple problem, from new knowledge to old knowledge, from proposition to proposition, from number to shape, from space to plane, from high dimension to low dimension, from multivariate to unary, from function to equation. The thinking method of reduction is the most basic thinking method in mathematics. The solution of all problems in mathematics is inseparable from civilization, and the idea of combining numbers and shapes embodies the mutual transformation between numbers and shapes; The idea of function and equation embodies the mutual transformation between function, equation and inequality. The idea of classified discussion embodies the mutual transformation of part and whole, and the above three ways of thinking are the concrete embodiment of the idea of reduction. Various transformation methods, analysis methods, reduction to absurdity, undetermined coefficient methods and construction methods are all means of transformation. Therefore, reduction is the soul of mathematical thinking method. )
Transformation includes equivalent transformation and non-equivalent transformation. Before and after equivalent transformation is a necessary and sufficient condition, so make the transformation as equivalent as possible; If necessary, restrictions should be attached to the unequal conversion to maintain equivalence or verify the conclusions.
Mastering basic knowledge, skills and methods skillfully and solidly is the basis of transformation; Rich association, astute and subtle observation, comparison and analogy are the bridges to realize transformation; To cultivate and train one's transformation consciousness requires a profound understanding of theorems, formulas and laws, a summary and refinement of typical exercises, and an active and conscious effort to discover the essential relationship between things. Some people think? Grasp the foundation and then transform it? It is the golden key to learn middle school mathematics well, which makes sense.
Common mathematical thinking method: combination of number and shape
The object of mathematical research is quantitative relationship and spatial form, that is? Count? With what? Shape? Two aspects. ? Count? With what? Shape? The two are not isolated, but closely related. The study of quantitative relationship can be transformed into the study of graphic properties, and conversely, the study of graphic properties can be transformed into the study of quantitative relationship, which is in the process of solving mathematical problems? Count? With what? Shape? The research strategy of interconversion is the idea of combining numbers with shapes.
The idea of the combination of numbers and shapes inspires people with the intuitive expression and profound and accurate quantitative expression of mathematical objects in almost all mathematical knowledge, and provides a simple and quick method to solve problems. Does its application often show? Another village? The harmonious and perfect combination of number and shape. Mr. Hua once had an incisive exposition: number and shape are interdependent, how can they be divided into two? Less is not so intuitive, and less is hard to be implicit. The combination of numbers and shapes is good in all aspects and wrong in all aspects. Don't forget that the unity of geometry and algebra is always inseparable. ?
The combination of numbers and shapes is not only an important mathematical idea, but also a common problem-solving strategy. On the one hand, many abstract concepts and analytical formulas of quantitative relations, if given geometric meaning, often become very intuitive; On the other hand, by studying the quantitative relationship, the properties of some graphs can be enriched, more accurate and more profound. This kind? Count? With what? Shape? The mutual transformation and infiltration of the two concepts can not only make the solution of some problems simple and vivid, but also greatly expand our thinking of solving problems. It can be said that the combination of numbers and shapes is not just about exploring ideas. Eyes? And a powerful way to deepen thinking? Lever? .
By who? Shape? Arrive? Count? This change is often more obvious, and by? Count? Arrive? Shape? Transformation needs the consciousness of transformation. Therefore, the application of the combination of numbers and shapes is often biased towards the reasons. Count? Arrive? Shape? Transformation.
In the college entrance examination, the characteristics of multiple-choice questions and fill-in-the-blank questions (only writing the results without writing the process) provide convenience for examining the idea of combining numbers and shapes, and can highlight the examinee's consciousness of transforming complex quantitative relations into intuitive geometric problems to solve. In solving problems, considering the rigor of reasoning and argumentation, the study of quantitative relations still highlights algebraic methods rather than advocating geometric methods. What is the reason for examining the combination of numbers and shapes in solving problems? Count? Arrive? Shape? Transformation is the mainstay.