Mathematical thinking, problem solving equation thinking
Mathematical thought has a higher level and status than basic knowledge of mathematics. It is contained in the process of the occurrence, development and application of mathematical knowledge, and 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.
Each mathematical thought and method has its own specific environment and basic theory, so that students can know what problems a mathematical thought is effective in solving, thus cultivating and improving students' ability to analyze and solve problems reasonably and correctly. The author now talks about some superficial understanding of the commonly used mathematical ideas in junior high school mathematics.
First, the application of equation thought
The so-called equation idea is to start with the analysis of the quantitative relationship of the problem, set the unknown quantity, transform the quantitative relationship between the known quantity and the unknown quantity in the problem into an equation or a mathematical model such as an equation, and then use the theory or method of the equation to solve the problem. It is clear, flexible and simple to analyze and deal with problems with equation thought.
The core of using equation is to reveal the quantitative relationship implied in the topic, set the unknown, construct the equation, and communicate the relationship between the known and the unknown, so as to solve the problem.
Second, the ideological application of the combination of numbers and shapes
Mathematics is a science that studies the relationship between spatial form and quantity in the real world. "Number" and "shape" are two basic concepts in mathematics, and each geometric figure contains a certain quantitative relationship; The quantitative relationship can often be intuitively reflected and described by geometric figures, so the combination of numbers and shapes has become an important thinking method for studying mathematical problems. In other words, both teachers and students should devote themselves to teaching activities. Student participation is particularly important. Without the active participation of students, such teaching activities will never succeed. For example, theorem teaching is the focus of mathematics teaching. How to let students discover the forming process of theorems, the origin of theorem proving thinking, especially the method of adding auxiliary lines, has always been the focus of teaching research.
In the teaching of "triangle midline theorem", we use computer-aided teaching means and "geometry sketchpad" software to create an ideal situation for students. The triangle drawn can be changed at will. (Theorem holds for any triangle) It can be calculated that a set of congruence angles are always equal, and the length of the middle line is half the length of the third side. Students can easily get the conclusion of the theorem by observing the graph. The essence of theorem proving is to transform a triangle figure into a parallelogram through translation transformation or rotation transformation. (Geometer's Sketchpad) can demonstrate the above process well. So it is natural to prove the theorem with auxiliary lines. Under the guidance of teachers, students actively participate, and the whole teaching process is a step-by-step process of students' thinking, which has achieved ideal teaching results.
The idea of the combination of numbers and shapes is the idea of combining the quantitative relationship and spatial form in the problem. In the method of solving problems, "number" and "shape" are transformed into each other, thus making the problem difficult to simplify and achieving the purpose of solving problems. The application of the combination of numbers and shapes can be divided into two situations: one is to clarify some properties of shapes with the help of the accuracy of numbers, that is, "to discuss shapes by numbers"; The other is to express a certain relationship between numbers with the help of geometric intuition of shapes, that is, "promoting numbers by shapes" uses the idea of combining numbers with shapes to solve problems, which makes it easy to find solutions, avoid complicated calculation and reasoning, and simplify the problem-solving process. Hua, a famous mathematician in China, once said: "If there are few shapes, it is less intuitive, and if there are few shapes, it is difficult to be nuanced;" Numbers and shapes are well combined, and everything is separated. "In mathematics, number and shape are two main research objects, and they are closely related. Under certain conditions, numbers and shapes can be transformed and infiltrated with each other.
Third, the application of classified discussion ideas
The idea of classified discussion is to divide the research object of mathematics into different kinds of mathematical ideas according to the similarities and differences of the essential attributes of mathematics. Correct application of the idea of classification is the basis of complete problem solving. For example, after learning the comparative magnitude of angles, angles smaller than a right angle can be divided into three categories: acute angle, right angle and obtuse angle, which is the embodiment of the classification idea. The same thing can be classified according to different standards, but under the same standard, it must be neither heavy nor leaking.
It is actually a strategy of "divide and rule" to divide the research object of a mathematical problem into several parts or situations according to certain standards, break it down into several parts and solve them one by one. The steps are as follows:
1. Determine the classification object-understand the classification concept;
2. Appropriate and reasonable classification-master the principle of classification;
3. Step-by-step discussion-learning classification methods;
4. Comprehensive summary-cultivate logical thinking.
The principles of classified discussion are: determining the object and unifying the standard; Grading, not leapfrog; No repetition, no omission.
The mathematical problem of classified discussion thinking occupies an important position in the process of mathematics learning because it has obvious logical characteristics and can train the order and generality of one's thinking.
Fourthly, the application of transformation and regression thought.
Complex problems are transformed into simple problems to solve, and unknown problems are transformed into known problems to solve ... Mathematical problems are often solved in constant transformation. The same mathematical problem, due to different observation angles and different levels of analysis and understanding, can lead to different goals and different solutions, but the purpose is only one-to simplify the complex as much as possible, simplify the difficult, make the unknown known, specialize the general and concretize the abstract.
Transformation includes equivalent transformation and non-equivalent transformation. Equivalent transformation requires that the cause and effect in the transformation process can be deduced from each other. But in fact, not all transformations are equivalent. So in the process of transformation, we must pay attention to the equivalence before and after transformation. If there are unequal transformations, additional constraints are required.
In a word, mathematical thought reflects the connection and essence of mathematical concepts, principles and laws, is a bridge to form mathematical ability and consciousness, and is the key to flexibly use mathematical knowledge, skills and methods. Mathematical thinking method is one of the important contents of middle school mathematics teaching. The solution of any mathematical problem is guided by mathematical thought and means by mathematical methods. Mathematics thought is the soul of teaching material system, the guide of teaching design, the commander-in-chief of classroom teaching and the guide of problem-solving thinking. It is an important measure to strengthen mathematics quality education to bring the essence of mathematics knowledge-mathematical thinking method into the category of basic knowledge. With the deepening of the teaching research of mathematical thinking method, the infiltration and implementation of mathematical thinking method in teaching will further improve the quality of mathematics teaching.