I. Observation
Observation is the most basic way for people to know things, the premise of finding and solving problems, and the starting point of thinking.
Mathematics is a science. To enable students to master rigorous learning methods, we must first cultivate their observation ability. Students' cognitive activities generally begin with their perception of practical things. Visual perception is often more important than hearing, and visual perception depends on whether students observe or not. To accept new knowledge or solve mathematical problems, we should start with observation, analyze while watching, and adopt different observation methods for different contents.
1. Direct investigation
Direct observation can rely on teaching AIDS or examples around you, demonstrate teaching AIDS or observe on the spot. Teach students how to observe, inspire and mobilize their initiative, and let them master the essence of knowledge by themselves.
2. Indirect observation
It can be applied to abstract concepts and laws, and through the combination of numbers and shapes, concepts or laws can be deduced from things and examples that students are familiar with. For example, when I talk about the properties of exponential function y=ax, I guide students to observe the characteristics and trends of graphics by combining images with numbers and shapes, find out the nature and scope of numbers, and students can sum up the properties. This will not make the learning process seem boring, but also improve students' observation and analysis ability.
3. Comparative observation
Comparative observation is mainly used to identify the similarities and differences of things. There will always be some similarities and connections in the same type of knowledge. Through comparative observation, students can deeply understand and strengthen concepts. Explain "Are two straight lines perpendicular to the same line, two straight lines perpendicular to the same plane, and two planes perpendicular to the same line parallel?" Let students compare and observe, grasp the similarities and differences in essence, avoid confusion and deepen understanding. By comparing various solutions to a problem, students can find the connection, look at the problem from different angles and cultivate their creative ability.
Second, Lenovo
Association is another way to think of another thing from one thing. The knowledge of each part of mathematics is interrelated and permeated with each other. In teaching, we should adopt various contact ways to cultivate students' thinking mode of "from one to two" and contact relevant knowledge.
1. Causal association
The relationship between knowledge and causality. When you see a right triangle, you will think of Pythagorean theorem, the sum of two acute angles is 90, projective theorem and so on.
2. Reversible association
Correlation can be one-way or two-way, and two-way correlation is reversible. There are many reversible processes in mathematics, so reversible association is often used. For example, the tangent formula: tan(α+β)=■, students should not only grasp the right from the left, but also explicitly push the left from the right, and make the following modifications to understand tanα+tanβ =( 1-tanα? Tanβ)tan(α+β), or tanα? Tanβ= 1- ■, when α+β = 45, it can be changed to tanα+tanβ= 1- tanα? Tanβ, that is, tanα+tanβ-tanα? Tanβ= 1。 If you add 1 to both sides at the same time, it can be written as (1+tan α) (1+tan β) = 2, so that students can understand and master the essence from different angles, learn the living formula, and freely associate with the same type of proof.
Third, explore
Exploration is a way for people to seek answers and solve problems in life, study and scientific research. The generality of mathematics knowledge is high, so it is very important for students to deeply understand, actively explore and be flexible.
Everything is interrelated and restricted, and mathematics also has this law. Spatial quadrilateral is a typical spatial model, from which many problems are derived. If you connect the midpoints of four sides in turn, what figure will you get? On this basis, add conditions:
1. If the diagonals are equal, what graph will you get?
2. If the diagonal is vertical, what figure will you get?
3. If the diagonals are equal and vertical, what figure will you get?
Different results can be obtained by changing the conditions in this way, which is conducive to cultivating students' divergent thinking and forming the habit of being diligent in thinking, good at exploring, giving inferences by analogy and solving problems flexibly.
Of course, while guiding students to explore independently, students should pay attention to the following aspects:
1. What keywords and concepts have been learned in the topic and how to understand them;
2. Excavate the connotation and extension of the concept;
3. Distinguish concepts with similar meanings and clarify similarities and differences. Let students explore effectively.
Fourth, transformation.
Contradictory transformation is one of the basic ideas of dialectics, which is often used in mathematics learning. When it is difficult for us to deal with mathematical problems directly, we can transform the problems through appropriate forms. Cultivate students' divergent dialectical thinking ability and creative thinking ability.
1. Grasp the transformation of structural characteristics
Some mathematical problems often have special forms. As long as the structural characteristics are analyzed, they can be flexibly transformed. If solving equation 2x-2=2x-2, the essence is the concept of disguised test: the absolute value of non-negative number is itself. If we master this characteristic, we can get x≥ 1.
2. Add auxiliary condition conversion
This method is often used in geometric proof. If it is proved that the public side of an isosceles triangle with a public side is perpendicular to the opposite side. It can be solved by heightening the auxiliary line on one side.
3. Use number and shape transformations
The thinking method of combining numbers and shapes is an effective method commonly used in mathematics, which can simplify more complicated problems. For example, it is proved that "the sum of squares of two diagonals of a parallelogram is equal to the sum of squares of two groups of opposite sides." With the help of vectors, it is easy to prove the quantitative relationship.
4. Transform the propositional goal
Some problems can be simplified by changing their characteristics into another form. For example, "finding the symmetry point of point (1, 2) about straight line y=0" is simply converted into "finding the symmetry point of point (1, 2) about x axis", which is easy to solve the problem.
In short, to cultivate and develop students' thinking ability, teachers should constantly guide students to observe, associate, explore and transform by means of comparison, analysis, reasoning and induction, think about problems in an orderly and hierarchical way, and analyze problems from the internal relations, movement changes and unity of opposites of things. Demonstrate to students, proceed from the reality of students, and strive to mobilize students' consciousness and initiative, so that students' thinking ability can be fully developed and improved.