1. Corresponding ideas and methods:
In the introductory teaching of algebra in senior one, there are calculated values of algebra evaluation. Through calculation, it is found that the value of algebraic expression is determined by the value of letters in algebraic expression, and different letter values can get different calculation results. Here, there is a correspondence between the value of letters and the value of algebra, then there is a correspondence between real numbers and points on the number axis, and there is a correspondence between ordered real numbers and points on the coordinate plane ... In this teaching design, attention should be paid to infiltrating corresponding ideas, which not only helps students to see problems from a changing point of view, but also helps to cultivate their functional concepts.
2. The idea and method of combining numbers and shapes.
The idea of combining numbers and shapes refers to a thinking strategy that combines numbers and shapes to analyze, study and solve problems. Mr. Hua, a famous mathematician, said: "Numbers and shapes are interdependent, how can they be divided into two?" If there are fewer numbers, there will be less intuition, and if there are fewer shapes, it will be difficult to be meticulous. The combination of numbers and shapes is good, and everything is separated. " This fully shows the importance of the combination of numbers and shapes in mathematical research and application.
(1) number thinking is combined with number and shape to solve the problem of number with shape.
For example, in the teaching of rational numbers and their operations, the concept of "quantity with opposite meanings" is consolidated by using the figure of "number axis", the concepts of reciprocal and absolute value are understood, the reasons for the size of rational numbers are grasped, the significance of addition and multiplication of rational numbers is understood, and the algorithm is mastered. In fact, for students, only through the combination of numbers and shapes can they better complete the learning tasks in this chapter. In addition, drawing a schematic diagram in the application of solving a series of equations in "one-dimensional linear equation" often brings ideas to solving problems. Chapter 9, the data in life, the choice of statistical chart and the review of statistical chart, shows the data with graphics, which is very intuitive and clear.
(2) Think about numbers with shapes, combine numbers with shapes, and solve numbers with shapes. For example, in the fourth chapter, "Plane Figure and its positional relationship", the length of line segment is expressed in terms of quantity, and the degree of angle is expressed in terms of quantity. Comparison of quantities is used to compare line segments and angles.
3. Overall ideas and methods
Holistic thinking means that when considering a mathematical problem, we should not focus on its local characteristics, but on the overall structure of the problem, understand the essence of the problem from a macro perspective through comprehensive and profound observation, and treat some independent but closely related quantities as a whole. Holistic thinking is widely used to deal with mathematical problems.
4. The idea and method of classification
There are many examples of classification in the textbook, such as rational number, real number, triangle, quadrilateral, etc., which can not only let students know the importance of classification: First, make related concepts systematic and complete; The second is to make the extension of the concept of classification clearer, deeper and more specific, and also to enable students to grasp the main points of the score: (1) classification is carried out according to certain standards, and the classification results are different with different standards; (2) It should be noted that the classification results are neither missing nor overlapping; (3) The classification should be gradual, and real numbers should not be divided into integers, fractions and irrational numbers.
5. Ideas and methods of analogy and association
When considering some problems, mathematics teaching design often puts forward assumptions and conjectures based on the similarity between things, thus extending the attribute analogy of known things to similar new things and promoting the discovery of new conclusions. For example, various algorithms of fractions are similar to those learned in primary school; For another example, the basic properties of the equation are obtained from the equilibrium condition ratio of the balance. This method embodies the teaching design principles of "knowing the new with law" and "introducing the new with the old". This design has a low starting point and is more easily accepted by students. Teaching provides background materials for thinking, which not only enlivens the classroom atmosphere, but also helps to complete the learning of new knowledge in a harmonious and relaxed atmosphere.
6. The method of reverse thinking
The so-called reverse thinking is to turn the problem upside down or think from the opposite side of the problem or use some mathematical formulas and laws to solve the problem. Strengthening the training of reverse thinking can cultivate the flexibility and divergence of students' thinking, so that students can effectively transfer their own mathematical knowledge, such as the number of absolute values equal to 2, the number of squares equal to 4, the number of cubes equal to 6, the reverse application after learning the powers of absolute values and rational numbers, and the reverse application of distribution laws.
7. Ideas and methods of transformation and transformation
Transformation consciousness refers to the transformation of problems in the process of solving problems, making them simple and familiar with the basic problem-solving mode. It is the idea and method of transforming one mathematical object into another under certain conditions. For example, using the concept of inverse number, the subtraction operation of rational number is transformed into addition operation; When learning equations and equations, "multivariate" can be transformed into "unitary" and "high order" can be transformed into "low order" equations to solve. It is the application of reduction idea to transform the inner angle of polygon into the inner angle of triangle and study it. They all adopt the problem-solving methods of turning the unknown into the known, the unfamiliar into the familiar and the complex into the simple. Their core is to transform the problems that have been solved into problems that have been clearly solved, so as to use the existing theories and technologies to deal with them.