I. Composition of the ability to analyze and solve problems
1. Ability to review problems
Examination of questions is a comprehensive understanding of conditions and problems, and an analysis of all situations related to conditions and problems. Is the premise of how to analyze and solve problems. Examining questions mainly refers to being able to fully understand the meaning of the question and grasp the essence of the question. The ability to analyze and discover implicit conditions, and the ability to simplify and transform what is known and sought. In order to solve problems quickly and accurately, it is very important to master the mathematical characteristics, transformation conditions or what to seek, and to find hidden conditions.
2. The ability to reasonably apply knowledge, ideas and methods to solve problems.
Senior high school mathematics knowledge includes functions, inequalities, series, trigonometric functions, complex numbers, solid geometry, analytic geometry and so on.
Capacity; Mathematical thoughts include the combination of numbers and shapes, the thought of function equation, classification discussion and equivalent transformation. Mathematical methods include undetermined coefficient method, method of substitution method, mathematical induction method, reduction to absurdity method, collocation method and other basic methods. Only by understanding and mastering the basic knowledge, ideas and methods of mathematics can we solve some basic problems in senior high school mathematics, and by selecting and applying knowledge, ideas and methods reasonably, we can solve problems faster and more smoothly.
2. Mathematical modeling ability
In recent years, there are several practical problems in the mathematics examination paper of college entrance examination, which challenge students' ability to analyze and solve problems, and mathematical modeling ability is an important way and core to solve practical problems.
Second, the strategy of cultivating and improving the ability to analyze and solve problems
1. Attach importance to the teaching of general methods and guide students to summarize and understand commonly used mathematical ideas and methods.
Compared with the basic knowledge of mathematics, mathematical thought has a higher level and status. It is included in the process of the occurrence, development and application of mathematical knowledge. It is a kind of mathematical consciousness, which belongs to the category of thinking and is used to understand, deal with and solve mathematical problems. Mathematical method is the concrete embodiment of mathematical thought, which has the characteristics of model and operability and can be used as a concrete means to solve problems. Only by summarizing mathematical ideas and methods can we be handy in analyzing and solving problems. Only by understanding the ideas and methods of mathematics can books and other people's knowledge and skills become their own abilities.
Every mathematical thought and method has its specific environment and basic theory. For example, the idea of classification discussion can be divided into: (1) the classification of common ratio in the summation formula of equal proportion series and the classification of slope in linear equation; (2) The contents that need to be classified in the deformation of the same solution, such as the discussion of parameters in parameter problems and the discussion of solution sets in solving inequality groups. Another example is the choice of mathematical methods, the commonly used collocation method for quadratic function problems, and the undetermined coefficient method for parameter problems. Therefore, in mathematics classroom teaching, we should attach importance to general methods, downplay special skills, and let students know the personality of a "thought" or "method", that is, know a mathematical thought or method.
2. Strengthen the teaching of application problems and improve students' pattern recognition ability.
College entrance examination is a ability-oriented examination, especially the ability of students to analyze and solve problems by using mathematical knowledge and methods, which is the focus of examination. The application questions in the college entrance examination focus on this ability, which can be seen from the difference in ability requirements between the test instructions in the new curriculum standard and the original test instructions. (The new curriculum version changes "the ability to analyze and solve problems" to "the ability to solve practical problems")
Mathematics is full of patterns. As far as solving practical problems is concerned, the understanding of its mathematical model is the premise of solving problems. Because the college entrance examination does not examine the original practical problems, the proposer designs and handles the original problems in production and life, so that each application problem has its own mathematical model. For example, the "transportation cost problem" in 1997 is a function and mean inequality; 1998 "sewage pool problem" is a function, several points and mean inequality; 1999' s "sparsity problem" is series, inequality and equation; The "tomato problem" in 2000 was piecewise linear function and quadratic function. In high school mathematics teaching, we should not only attach importance to the teaching of applied problems, but also carry out special training on applied problems, so as to guide students to summarize and summarize mathematical models of various applied problems, so that students can use mathematical ideas and methods reasonably to analyze and solve practical problems.
3. Properly train open questions and new questions to broaden students' knowledge.
To analyze and solve a problem, we must first understand the meaning of the problem, and then we can further use mathematical ideas and methods to solve the problem. In recent years, with the rapid development of new technological revolution, mathematics education is required to cultivate talents with high mathematics quality and strong innovation ability. This is reflected in the emergence of some new background questions and open questions in the college entrance examination, which pay more attention to the examination of ability. Because the open questions are characterized by insufficient conditions or inconclusive conclusions, the background of the new background questions is new, which causes great trouble for students to understand the meaning of the questions and choose the solution methods, resulting in a high loss rate, such as 1999 science questions 16 and 22, a lot.
Students are at a loss because they don't understand "ridge" and "reduction rate does not exceed"; Another example is the 2000 liberal arts 16 and 2 1 questions, and the 2006 spring college entrance examination 1 1 questions. Only by understanding the given figures can you answer correctly. Therefore, in senior high school mathematics teaching, it is necessary to properly train open questions and new types of questions to broaden students' knowledge.