First, the improvement of spatial imagination.
At the beginning of learning, students should first look at simple solid geometry topics, not start with difficult problems. Draw some three-dimensional geometric figures by yourself, such as exercises in textbooks and exercises in tutorials. Draw it yourself without looking at the original picture. It is a good thing that the number drawn may be different from the given number. By contrast, that number is easier to solve the problem.
Second, the cultivation of logical thinking ability
To cultivate logical thinking ability, we must first firmly grasp the basic knowledge of mathematics, and then master the necessary logical knowledge and logical thinking.
1. Strengthen the understanding of basic concepts
Mathematical concept is one of the two components of mathematical knowledge system, and understanding and mastering mathematical concept is the key to learn mathematics well and improve mathematical ability.
To understand the basic concepts, we must first think more. For example, the understanding of straight lines on different planes is a simple definition. How can it not be in the same plane? The first one is a straight line leaving the same plane from this plane, or drawing with two pens, which gives us an intuitive concept of straight lines in different planes, and then we are thinking about how to mathematically ensure that two straight lines are not in the same plane, and those conditions can ensure that two straight lines are not in the same plane. If you think about it more, you will know that as long as straight lines are not parallel and intersect, they are different. For the case of non-parallelism, we already know how to ensure non-intersection in plane geometry and whether the extension line can be proved. If not, then put one of the straight lines on a plane and see if the other straight line is parallel to this plane, so that we can easily grasp the concept of non-planar straight line.
2. Strengthen the understanding of mathematical propositions and learn to use mathematical propositions flexibly to solve problems.
The understanding and application of mathematical axioms and theorems are prominent in the proof and calculation of topics. Students need to avoid the use of imprecise logical reasoning, unconfirmed theorems, axioms and rules, or subjective assumptions instead of rigorous scientific argumentation, unreasonable writing format, unclear levels, improper use of mathematical symbol language and unaccustomed.
(1) attaches importance to the proof of the theorem itself. We know that the proof idea of theorem itself is exemplary and typical, which embodies the cultivation of basic logical reasoning knowledge and basic proof idea, as well as the cultivation of standardized writing format. We can not only analyze the conditions and conclusions of the theorem, but also master the content of the theorem, the thinking method of proof, the scope of application and the expression form. Especially after entering high school, it involves some new ways of thinking, such as the example of solid geometry in the new textbook: "A straight line passes through a point outside the plane and a point in the plane, and all straight lines in the plane that do not pass through this point are non-planar straight lines." The proof of this theorem adopts the method of reduction to absurdity, so the proof idea of reduction to absurdity needs to be understood, general steps, writing format, matters needing attention and so on. And after proper training, we can master the application of reduction to absurdity in proving solid geometry problems.
(2) Improve the ability of applying theorem to analyze and solve problems. For exercises, we must first know what to do (what is the required conclusion) and what conditions can meet the requirements, so that we can find the conditions step by step. Of course, it depends on the specific situation and needs more exercise. Necessary exercises are essential.